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SPC product codes under the BEC S. D. Cardell, J. J. Climent, Performance of SPC product codes under the erasure A. Lpez Martn channel Sara D. Cardell 1 Joan-Josep Climent 1 Alberto Lpez Martn 2 1 Universitat dAlacant, Spain 2

  1. SPC product code SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López ◮ C − = C � is a linear binary code with parameters [ n , n − 1 , 2]. Martín ◮ We consider the product code C = C − ⊗ C � . Prelimina- ries ◮ The parameters of the product code are [ n 2 , ( n − 1) 2 , 4]. SPC product code Kotska ◮ The code C corrects only 3 erasures. Numbers Counting patterns In some special cases this code can correct more than 3 erasures. Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  2. SPC product code SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López ◮ C − = C � is a linear binary code with parameters [ n , n − 1 , 2]. Martín ◮ We consider the product code C = C − ⊗ C � . Prelimina- ries ◮ The parameters of the product code are [ n 2 , ( n − 1) 2 , 4]. SPC product code Kotska ◮ The code C corrects only 3 erasures. Numbers Counting patterns In some special cases this code can correct more than 3 erasures. Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  3. SPC product code SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López ◮ C − = C � is a linear binary code with parameters [ n , n − 1 , 2]. Martín ◮ We consider the product code C = C − ⊗ C � . Prelimina- ries ◮ The parameters of the product code are [ n 2 , ( n − 1) 2 , 4]. SPC product code Kotska ◮ The code C corrects only 3 erasures. Numbers Counting patterns In some special cases this code can correct more than 3 erasures. Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  4. Erasure pattern SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín Definition Prelimina- ries An erasure pattern of size m × m , with t erasures, where 0 ≤ t ≤ m 2 and SPC product code 1 ≤ m ≤ n , is an array of size m × m where t of the entries correspond to Kotska Numbers the position of the erasures. Counting patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  5. Decoding SPC product codes under the BEC S. D. Cardell, J. J. Climent, ◮ An erasure pattern of size n × n corresponds to words of size n × n , A. López Martín where the position of the erasures is the unique information we consider. Prelimina- ries ◮ Given a received word with t erasures, the decoder will perform iterative SPC product code row-wise and column-wise decoding to recover the erased bits. Kotska Numbers ◮ When a single bit is erased in a row or column, it can be recovered. Counting patterns ◮ If more than one bit is erased in a row (column), it is skipped. Conclusions ◮ Decoding is performed until no further recovery is possible. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  6. Decoding SPC product codes under the BEC S. D. Cardell, J. J. Climent, ◮ An erasure pattern of size n × n corresponds to words of size n × n , A. López Martín where the position of the erasures is the unique information we consider. Prelimina- ries ◮ Given a received word with t erasures, the decoder will perform iterative SPC product code row-wise and column-wise decoding to recover the erased bits. Kotska Numbers ◮ When a single bit is erased in a row or column, it can be recovered. Counting patterns ◮ If more than one bit is erased in a row (column), it is skipped. Conclusions ◮ Decoding is performed until no further recovery is possible. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  7. Decoding SPC product codes under the BEC S. D. Cardell, J. J. Climent, ◮ An erasure pattern of size n × n corresponds to words of size n × n , A. López Martín where the position of the erasures is the unique information we consider. Prelimina- ries ◮ Given a received word with t erasures, the decoder will perform iterative SPC product code row-wise and column-wise decoding to recover the erased bits. Kotska Numbers ◮ When a single bit is erased in a row or column, it can be recovered. Counting patterns ◮ If more than one bit is erased in a row (column), it is skipped. Conclusions ◮ Decoding is performed until no further recovery is possible. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  8. Decoding SPC product codes under the BEC S. D. Cardell, J. J. Climent, ◮ An erasure pattern of size n × n corresponds to words of size n × n , A. López Martín where the position of the erasures is the unique information we consider. Prelimina- ries ◮ Given a received word with t erasures, the decoder will perform iterative SPC product code row-wise and column-wise decoding to recover the erased bits. Kotska Numbers ◮ When a single bit is erased in a row or column, it can be recovered. Counting patterns ◮ If more than one bit is erased in a row (column), it is skipped. Conclusions ◮ Decoding is performed until no further recovery is possible. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  9. Decoding SPC product codes under the BEC S. D. Cardell, J. J. Climent, ◮ An erasure pattern of size n × n corresponds to words of size n × n , A. López Martín where the position of the erasures is the unique information we consider. Prelimina- ries ◮ Given a received word with t erasures, the decoder will perform iterative SPC product code row-wise and column-wise decoding to recover the erased bits. Kotska Numbers ◮ When a single bit is erased in a row or column, it can be recovered. Counting patterns ◮ If more than one bit is erased in a row (column), it is skipped. Conclusions ◮ Decoding is performed until no further recovery is possible. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  10. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  11. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  12. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- × ries SPC product code × × Kotska Numbers × Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  13. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- × ries SPC product code × × Kotska Numbers × Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  14. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers × Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  15. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers × Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  16. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers × Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  17. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  18. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  19. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × × Conclusions × × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  20. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × × Conclusions × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  21. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × × Conclusions × ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  22. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  23. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  24. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  25. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × × Kotska Numbers Counting patterns × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  26. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × Kotska Numbers Counting patterns × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  27. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × Kotska Numbers Counting patterns × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  28. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code × Kotska Numbers Counting patterns × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  29. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  30. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns × Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  31. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  32. Decoding SPC product Example (cont.) codes under the Consider the SPC code � BEC C with parameters [6 , 5 , 2]. S. D. We can construct the binary product code C = � C ⊗ � C with parameters Cardell, J. J. Climent, [36 , 25 , 4]. A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  33. Decoding SPC product codes under the Definition BEC S. D. An erasure pattern of size m × m is said to be uncorrectable if and only if it Cardell, J. J. Climent, contains a subpattern of size l × l , l ≤ m , such that each row and each A. López column have two or more erasures. Martín Prelimina- ries Example SPC product code Kotska × × × × × × Numbers Counting × × × patterns Conclusions × × × × × (a) Correctable erasure pattern of (b) Uncorrectable erasure pattern size 4 × 4 with 7 erasures of size 4 × 4 with 7 erasures S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  34. Erasure pattern SPC product codes under the BEC S. D. Cardell, J. Example J. Climent, A. López Martín × × Prelimina- ries × × SPC product code × × Kotska Numbers Counting × × patterns Conclusions Figure: Uncorrectable erasure pattern of size 4 × 4 with 7 erasures S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  35. Erasure pattern SPC product codes under the BEC S. D. Cardell, J. Example J. Climent, A. López Martín × × Prelimina- ries × × SPC product code × × Kotska Numbers Counting × × patterns Conclusions Figure: Uncorrectable erasure pattern of size 4 × 4 with 7 erasures S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  36. Decoding SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín ◮ Erasure patterns of size n × n with 3 erasures or less are always Prelimina- correctable. ries SPC product code Kotska ◮ Erasure patterns of size n × n with t erasures, 4 ≤ t ≤ 2 n − 1 may or Numbers Counting may not be correctable. patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  37. Decoding SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín ◮ Erasure patterns of size n × n with 3 erasures or less are always Prelimina- correctable. ries SPC product code Kotska ◮ Erasure patterns of size n × n with t erasures, 4 ≤ t ≤ 2 n − 1 may or Numbers Counting may not be correctable. patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  38. Decoding SPC product codes under the Example (cont.) BEC Consider the SPC code � S. D. C with parameters [6 , 5 , 2]. Cardell, J. J. Climent, We can construct the binary product code C = � C ⊗ � C with parameters A. López Martín [36 , 25 , 4]. Prelimina- ries SPC product code × × × × × × × × × × × Kotska Numbers × × × Counting patterns × × Conclusions × × × × × × S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  39. Classification SPC product codes under the BEC S. D. Cardell, J. Definition J. Climent, A. López Martín An uncorrectable erasure pattern is said to be strict if none of the erasures can be corrected. Equally, an uncorrectable erasure pattern is said to be Prelimina- ries partial if it can be partially corrected. SPC product code Kotska Numbers × × × × × × Counting patterns × × × × Conclusions × × × × S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  40. Strict uncorrectable erasure patterns SPC product codes under the BEC S. D. Cardell, J. Lemma J. Climent, A. López Martín An strict uncorrectable erasure pattern contains two or more erasures in each row and column in error. Prelimina- ries SPC product code Kotska × × × Numbers Counting × × patterns Conclusions × × S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  41. Partial uncorrectable erasure pattern SPC product codes under the BEC S. D. Cardell, J. Lemma J. Climent, A. López Martín A partial uncorrectable erasure pattern always contains an strict uncorrectable erasure pattern. Prelimina- ries SPC product code Kotska × × × Numbers Counting × × patterns Conclusions × × S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  42. Idea SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín Purpose Prelimina- ries We would like to count the number of uncorrectable erasure patterns of size SPC product n × n with t erasures, 4 ≤ t ≤ 2 n − 1. code Kotska Numbers Counting patterns In this work, we count the number of strict uncorrectable erasure patterns. Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  43. Idea SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín Purpose Prelimina- ries We would like to count the number of uncorrectable erasure patterns of size SPC product n × n with t erasures, 4 ≤ t ≤ 2 n − 1. code Kotska Numbers Counting patterns In this work, we count the number of strict uncorrectable erasure patterns. Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  44. Partition of an integer SPC product codes under the BEC Definition S. D. Cardell, J. J. Climent, If t is a positive integer, then a partition of t is a non-increasing sequence of positive integers ( λ 1 , λ 2 , λ 3 , . . . , λ p ) such that � p A. López i =1 λ i = t . Martín Prelimina- ries SPC product We denote by P t the set of possible partitions of the integer t . code Kotska Numbers Counting Example patterns Conclusions For example, the set of partitions of 6 is given by P 6 = { (6) , (5 , 1) , (4 , 2) , (4 , 1 , 1) , (3 , 3) , (3 , 2 , 1) , (3 , 1 , 1 , 1) , (2 , 2 , 2) , (2 , 2 , 1 , 1) , (2 , 1 , 1 , 1 , 1) , (1 , 1 , 1 , 1 , 1 , 1) } . S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  45. Partition of an integer SPC product codes under the BEC Definition S. D. Cardell, J. J. Climent, If t is a positive integer, then a partition of t is a non-increasing sequence of positive integers ( λ 1 , λ 2 , λ 3 , . . . , λ p ) such that � p A. López i =1 λ i = t . Martín Prelimina- ries SPC product We denote by P t the set of possible partitions of the integer t . code Kotska Numbers Counting Example patterns Conclusions For example, the set of partitions of 6 is given by P 6 = { (6) , (5 , 1) , (4 , 2) , (4 , 1 , 1) , (3 , 3) , (3 , 2 , 1) , (3 , 1 , 1 , 1) , (2 , 2 , 2) , (2 , 2 , 1 , 1) , (2 , 1 , 1 , 1 , 1) , (1 , 1 , 1 , 1 , 1 , 1) } . S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  46. Partition of an integer SPC product codes under the BEC Definition S. D. Cardell, J. J. Climent, If t is a positive integer, then a partition of t is a non-increasing sequence of positive integers ( λ 1 , λ 2 , λ 3 , . . . , λ p ) such that � p A. López i =1 λ i = t . Martín Prelimina- ries SPC product We denote by P t the set of possible partitions of the integer t . code Kotska Numbers Counting Example patterns Conclusions For example, the set of partitions of 6 is given by P 6 = { (6) , (5 , 1) , (4 , 2) , (4 , 1 , 1) , (3 , 3) , (3 , 2 , 1) , (3 , 1 , 1 , 1) , (2 , 2 , 2) , (2 , 2 , 1 , 1) , (2 , 1 , 1 , 1 , 1) , (1 , 1 , 1 , 1 , 1 , 1) } . S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  47. Conjugate partition SPC product codes under the BEC S. D. Cardell, J. J. Climent, Definition A. López Martín Let us consider a partition λ = ( λ 1 , λ 2 , . . . , λ p ) of t . The conjugate of λ is defined as the vector λ ∗ = ( λ ∗ Prelimina- 1 , λ ∗ 2 , . . . , λ ∗ p ′ ) where ries SPC product code λ ∗ j = |{ i | 1 ≤ i ≤ p , λ i ≥ j }| . Kotska Numbers Counting patterns Conclusions Notice that both, λ and λ ∗ , are partitions of the same integer t . S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  48. Conjugate partition SPC product codes under the BEC S. D. Cardell, J. J. Climent, Definition A. López Martín Let us consider a partition λ = ( λ 1 , λ 2 , . . . , λ p ) of t . The conjugate of λ is defined as the vector λ ∗ = ( λ ∗ Prelimina- 1 , λ ∗ 2 , . . . , λ ∗ p ′ ) where ries SPC product code λ ∗ j = |{ i | 1 ≤ i ≤ p , λ i ≥ j }| . Kotska Numbers Counting patterns Conclusions Notice that both, λ and λ ∗ , are partitions of the same integer t . S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  49. Conjugate partition SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Example Martín λ = (3 , 1) and λ ∗ = (2 , 1 , 1) are conjugate partitions of t = 4. Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  50. Young diagram and Young tableaux SPC Let λ = ( λ 1 , λ 2 , . . . , λ p ) and µ = ( µ 1 , µ 2 , . . . , µ q ) be two partitions of the same product codes integer t . under the BEC S. D. Cardell, J. Definition J. Climent, A. López A Young diagram of shape λ is an arrangement of t boxes in p rows where Martín there are λ i boxes in row i , with i = 1 , 2 , . . . , p , and these boxes are left Prelimina- justified. ries SPC product code Kotska Numbers Counting patterns Conclusions λ = (4 , 3 , 2 , 2) S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  51. Young diagram and Young tableaux SPC Let λ = ( λ 1 , λ 2 , . . . , λ p ) and µ = ( µ 1 , µ 2 , . . . , µ q ) be two partitions of the same product codes integer t . under the BEC S. D. Cardell, J. Definition J. Climent, A. López A Young tableau of shape λ and content µ is obtained from a Young Martín diagram of shape λ by inserting in each box one of the integers 1 , 2 , . . . , q in Prelimina- such a way that the following conditions hold: ries SPC product i) the elements in each row are non-decreasing, code Kotska Numbers ii) the elements in each column are strictly increasing, Counting iii) the integer j occurs µ j times, with j = 1 , 2 , . . . , q . patterns Conclusions λ = (4 , 3 , 2 , 2) S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  52. Young diagram and Young tableaux SPC Let λ = ( λ 1 , λ 2 , . . . , λ p ) and µ = ( µ 1 , µ 2 , . . . , µ q ) be two partitions of the same product codes integer t . under the BEC S. D. Cardell, J. Definition J. Climent, A. López A Young tableau of shape λ and content µ is obtained from a Young Martín diagram of shape λ by inserting in each box one of the integers 1 , 2 , . . . , q in Prelimina- such a way that the following conditions hold: ries SPC product i) the elements in each row are non-decreasing, code Kotska Numbers ii) the elements in each column are strictly increasing, Counting iii) the integer j occurs µ j times, with j = 1 , 2 , . . . , q . patterns Conclusions 1 1 1 2 2 2 3 λ = (4 , 3 , 2 , 2) µ = (3 , 3 , 2 , 2 , 1) 3 4 4 5 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  53. Young diagram and Young tableaux SPC product codes This is not the only option; there are 5 more Young tableaux with these under the BEC properties. S. D. Cardell, J. J. Climent, A. López Martín 1 1 1 2 1 1 1 2 1 1 1 2 Prelimina- 5 2 2 3 2 2 4 2 2 ries SPC product 3 4 3 3 3 3 code Kotska Numbers 4 5 4 5 4 4 Counting patterns Conclusions 1 1 1 3 1 1 1 4 1 1 1 5 2 2 2 2 2 2 2 2 2 3 4 3 3 3 3 4 5 4 5 4 4 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  54. Kotska numbers SPC product Definition codes under the BEC If λ = ( λ 1 , λ 2 , . . . , λ p ) and µ = ( µ 1 , µ 2 , . . . , µ q ) are two partitions of the same S. D. integer t , then the Kostka number denoted by κ λ,µ , is the number of Young Cardell, J. J. Climent, tableaux of shape λ and content µ . A. López Martín Prelimina- ries Example SPC product code We want to compute κ (3 , 2 , 1) , (3 , 2 , 1) . Kotska Numbers We have to count the number of Young tableaux of shape λ = (3 , 2 , 1) and Counting content µ = (3 , 2 , 1). patterns Conclusions 1 1 1 2 2 3 Thus, κ (3 , 2 , 1) , (3 , 2 , 1) = 1. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  55. Kotska numbers SPC product Definition codes under the BEC If λ = ( λ 1 , λ 2 , . . . , λ p ) and µ = ( µ 1 , µ 2 , . . . , µ q ) are two partitions of the same S. D. integer t , then the Kostka number denoted by κ λ,µ , is the number of Young Cardell, J. J. Climent, tableaux of shape λ and content µ . A. López Martín Prelimina- ries Example SPC product code We want to compute κ (3 , 2 , 1) , (3 , 2 , 1) . Kotska Numbers We have to count the number of Young tableaux of shape λ = (3 , 2 , 1) and Counting content µ = (3 , 2 , 1). patterns Conclusions 1 1 1 2 2 3 Thus, κ (3 , 2 , 1) , (3 , 2 , 1) = 1. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  56. Kotska numbers SPC product Definition codes under the BEC If λ = ( λ 1 , λ 2 , . . . , λ p ) and µ = ( µ 1 , µ 2 , . . . , µ q ) are two partitions of the same S. D. integer t , then the Kostka number denoted by κ λ,µ , is the number of Young Cardell, J. J. Climent, tableaux of shape λ and content µ . A. López Martín Prelimina- ries Example SPC product code We want to compute κ (3 , 2 , 1) , (3 , 2 , 1) . Kotska Numbers We have to count the number of Young tableaux of shape λ = (3 , 2 , 1) and Counting content µ = (3 , 2 , 1). patterns Conclusions 1 1 1 2 2 3 Thus, κ (3 , 2 , 1) , (3 , 2 , 1) = 1. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  57. Kotska numbers SPC product Definition codes under the BEC If λ = ( λ 1 , λ 2 , . . . , λ p ) and µ = ( µ 1 , µ 2 , . . . , µ q ) are two partitions of the same S. D. integer t , then the Kostka number denoted by κ λ,µ , is the number of Young Cardell, J. J. Climent, tableaux of shape λ and content µ . A. López Martín Prelimina- ries Example SPC product code We want to compute κ (3 , 2 , 1) , (3 , 2 , 1) . Kotska Numbers We have to count the number of Young tableaux of shape λ = (3 , 2 , 1) and Counting content µ = (3 , 2 , 1). patterns Conclusions 1 1 1 2 2 3 Thus, κ (3 , 2 , 1) , (3 , 2 , 1) = 1. S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  58. Counting binary matrices SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín ◮ Let A be a binary matrix of size m 1 × m 2 . Prelimina- ◮ Let R = ( r 1 , r 2 , . . . , r m 1 ) be the vector where r i is the sum of the elements ries in row i of matrix A . SPC product code Kotska ◮ Let C = ( c 1 , c 2 , . . . , c m 2 ) be the vector where c j is the sum of the Numbers elements in row j of matrix A . Counting patterns ◮ Note that r 1 + r 2 + · · · + r m 1 = c 1 + c 2 + · · · + c m 2 and call this number t . Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  59. Counting binary matrices SPC product codes under the BEC S. D. Cardell, J. R. A. Brualdi, Algorithms for constructing (0,1)-matrices with prescribed J. Climent, A. López row and column sum vectors, Discrete Mathematics 306 (23) (2006) Martín 3054–3062. Prelimina- ries Theorem SPC product code Kotska The number of binary matrices with R and C as the row sum and the Numbers Counting column sum, respectively, is given by patterns � Conclusions κ λ, R κ λ ∗ , C . λ ∈P t S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  60. Counting binary matrices SPC product codes under the BEC S. D. Cardell, J. R. A. Brualdi, Algorithms for constructing (0,1)-matrices with prescribed J. Climent, A. López row and column sum vectors, Discrete Mathematics 306 (23) (2006) Martín 3054–3062. Prelimina- ries Theorem SPC product code Kotska The number of binary matrices with R and C as the row sum and the Numbers Counting column sum, respectively, is given by patterns � Conclusions κ λ, R κ λ ∗ , C . λ ∈P t S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  61. Outline SPC product codes under the BEC S. D. Cardell, J. J. Climent, 1 Preliminaries A. López Martín SPC product code Kotska Numbers Prelimina- ries SPC product code Kotska Numbers 2 Counting patterns Counting patterns Conclusions 3 Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  62. Counting patterns SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín Prelimina- Purpose ries SPC product We would like to count the number of uncorrectable erasure patterns of size code Kotska n × n with t erasures, 4 ≤ t ≤ 2 n − 1. Numbers Counting patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  63. Uncorrectable patterns with four erasures SPC product Assume we have a codeword of size n × n and that 4 erasures have codes under the occurred. BEC S. D. The only uncorrectable erasure pattern of 4 erasures is formed by a square: Cardell, J. J. Climent, A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 4 erasures is � n � 2 . 2 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  64. Uncorrectable patterns with four erasures SPC product Assume we have a codeword of size n × n and that 4 erasures have codes under the occurred. BEC S. D. The only uncorrectable erasure pattern of 4 erasures is formed by a square: Cardell, J. J. Climent, A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 4 erasures is � n � 2 . 2 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  65. Uncorrectable patterns with four erasures SPC product Assume we have a codeword of size n × n and that 4 erasures have codes under the occurred. BEC S. D. The only uncorrectable erasure pattern of 4 erasures is formed by a square: Cardell, J. J. Climent, A. López Martín Prelimina- × × ries SPC product × × code Kotska Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 4 erasures is � n � 2 . 2 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  66. Uncorrectable patterns with four erasures SPC product Assume we have a codeword of size n × n and that 4 erasures have codes under the occurred. BEC S. D. The only uncorrectable erasure pattern of 4 erasures is formed by a square: Cardell, J. J. Climent, A. López Martín Prelimina- × × ries SPC product code Kotska Numbers × × Counting patterns Conclusions The total number of uncorrectable erasure patterns with 4 erasures is � n � 2 . 2 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  67. Uncorrectable patterns with four erasures SPC product Assume we have a codeword of size n × n and that 4 erasures have codes under the occurred. BEC S. D. The only uncorrectable erasure pattern of 4 erasures is formed by a square: Cardell, J. J. Climent, A. López Martín Prelimina- × × ries SPC product code Kotska Numbers × × Counting patterns Conclusions The total number of uncorrectable erasure patterns with 4 erasures is � n � 2 . 2 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  68. Uncorrectable patterns with four erasures SPC product Assume we have a codeword of size n × n and that 4 erasures have codes under the occurred. BEC S. D. The only uncorrectable erasure pattern of 4 erasures is formed by a square: Cardell, J. J. Climent, A. López Martín Prelimina- × × ries SPC product code Kotska Numbers × × Counting patterns Conclusions The total number of uncorrectable erasure patterns with 4 erasures is � n � 2 . 2 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  69. Uncorrectable patterns with five erasures SPC product Assume we have a codeword of size n × n and that 5 erasures have codes under the occurred. BEC The only uncorrectable erasure pattern of 5 erasures is formed by a square S. D. Cardell, J. (uncorrectable pattern of size 2 × 2) and one extra erasure: J. Climent, A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 5 erasures is � n � 2 � n 2 − 4 � . 2 1 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  70. Uncorrectable patterns with five erasures SPC product Assume we have a codeword of size n × n and that 5 erasures have codes under the occurred. BEC The only uncorrectable erasure pattern of 5 erasures is formed by a square S. D. Cardell, J. (uncorrectable pattern of size 2 × 2) and one extra erasure: J. Climent, A. López Martín Prelimina- ries SPC product code Kotska Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 5 erasures is � n � 2 � n 2 − 4 � . 2 1 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  71. Uncorrectable patterns with five erasures SPC product Assume we have a codeword of size n × n and that 5 erasures have codes under the occurred. BEC The only uncorrectable erasure pattern of 5 erasures is formed by a square S. D. Cardell, J. (uncorrectable pattern of size 2 × 2) and one extra erasure: J. Climent, A. López Martín × × Prelimina- ries × × SPC product code Kotska × Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 5 erasures is � n � 2 � n 2 − 4 � . 2 1 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  72. Uncorrectable patterns with five erasures SPC product Assume we have a codeword of size n × n and that 5 erasures have codes under the occurred. BEC The only uncorrectable erasure pattern of 5 erasures is formed by a square S. D. Cardell, J. (uncorrectable pattern of size 2 × 2) and one extra erasure: J. Climent, A. López Martín × × Prelimina- ries × × SPC product code Kotska × Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 5 erasures is � n � 2 � n 2 − 4 � . 2 1 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  73. Uncorrectable patterns with five erasures SPC product Assume we have a codeword of size n × n and that 5 erasures have codes under the occurred. BEC The only uncorrectable erasure pattern of 5 erasures is formed by a square S. D. Cardell, J. (uncorrectable pattern of size 2 × 2) and one extra erasure: J. Climent, A. López Martín × × × Prelimina- ries × × SPC product code Kotska Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 5 erasures is � n � 2 � n 2 − 4 � . 2 1 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  74. Uncorrectable patterns with five erasures SPC product Assume we have a codeword of size n × n and that 5 erasures have codes under the occurred. BEC The only uncorrectable erasure pattern of 5 erasures is formed by a square S. D. Cardell, J. (uncorrectable pattern of size 2 × 2) and one extra erasure: J. Climent, A. López Martín × × × Prelimina- ries × × SPC product code Kotska Numbers Counting patterns Conclusions The total number of uncorrectable erasure patterns with 5 erasures is � n � 2 � n 2 − 4 � . 2 1 S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  75. Uncorrectable patterns with six erasures SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín Prelimina- ries The total number of uncorrectable erasure patterns with 6 erasures is � n � 2 � n 2 − 4 � − 4 � n �� n � + 6 � n � 2 . SPC product code Kotska 2 2 2 3 3 Numbers Counting patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  76. Erasure patterns/ binary matrices SPC product codes under the Let us represent an erasure pattern of size n × n by a binary matrix of size BEC S. D. n × n where there is 1 in the erasure positions and 0 otherwise. Cardell, J. J. Climent, A. López × ×   Martín 0 0 1 1 ×  1 0 0 0  Prelimina-   ries × 1 0 0 0 SPC product code 1 0 0 0 × Kotska Numbers Counting patterns Conclusions In this work, our purpose is to count the number of strict uncorrectable erasure patterns of size n × n with t erasures. Equivalently, we want to find all matrices of size n × n with t ones (and 2 or more ones in each non-zero row and non-zero column). S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  77. Erasure patterns/ binary matrices SPC product codes under the Let us represent an erasure pattern of size n × n by a binary matrix of size BEC S. D. n × n where there is 1 in the erasure positions and 0 otherwise. Cardell, J. J. Climent, A. López × ×   Martín 0 0 1 1 ×  1 0 0 0  Prelimina-   ries × 1 0 0 0 SPC product code 1 0 0 0 × Kotska Numbers Counting patterns Conclusions In this work, our purpose is to count the number of strict uncorrectable erasure patterns of size n × n with t erasures. Equivalently, we want to find all matrices of size n × n with t ones (and 2 or more ones in each non-zero row and non-zero column). S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  78. Uncorrectable strict erasure patterns SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín Notation Prelimina- ◮ The partitions that we will consider will have all length n . ries SPC product code ◮ If a partition has length r < n , it will be filled in with n − r zeros. Kotska Numbers ◮ For example, (6 , 1) is a partition of 7 with length 2, but if we are Counting considering partitions of length 4, we write (6 , 1 , 0 , 0). patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  79. Uncorrectable strict erasure patterns SPC product codes under the BEC S. D. Cardell, J. J. Climent, A. López Martín Notation Prelimina- ◮ The partitions that we will consider will have all length n . ries SPC product code ◮ If a partition has length r < n , it will be filled in with n − r zeros. Kotska Numbers ◮ For example, (6 , 1) is a partition of 7 with length 2, but if we are Counting considering partitions of length 4, we write (6 , 1 , 0 , 0). patterns Conclusions S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  80. Notation SPC product codes under the Definition BEC S. D. Cardell, J. Given two positive integers n and t , the set P n t is a subset of the set of J. Climent, A. López partitions of t of length n defined as Martín P n t = { λ ∈ P t | λ i � = 1 , λ i ≤ n , i = 1 , 2 , . . . , n } . Prelimina- ries SPC product code Kotska Example Numbers Counting Consider t = 6 and n = 3. patterns Conclusions P 3 6 = { (2 , 2 , 2) , (3 , 3 , 0) } S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  81. Notation SPC product codes under the Definition BEC S. D. Cardell, J. Given two positive integers n and t , the set P n t is a subset of the set of J. Climent, A. López partitions of t of length n defined as Martín P n t = { λ ∈ P t | λ i � = 1 , λ i ≤ n , i = 1 , 2 , . . . , n } . Prelimina- ries SPC product code Kotska Example Numbers Counting Consider t = 6 and n = 3. patterns Conclusions P 6 = { (6) , (5 , 1) , (4 , 2) , (4 , 1 , 1) , (3 , 3) , (3 , 2 , 1) , (3 , 1 , 1 , 1) , (2 , 2 , 2) , (2 , 2 , 1 , 1) , (2 , 1 , 1 , 1 , 1) , (1 , 1 , 1 , 1 , 1 , 1) } . P 3 6 = { (2 , 2 , 2) , (3 , 3 , 0) } S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  82. Notation SPC product codes under the Definition BEC S. D. Cardell, J. Given two positive integers n and t , the set P n t is a subset of the set of J. Climent, A. López partitions of t of length n defined as Martín P n t = { λ ∈ P t | λ i � = 1 , λ i ≤ n , i = 1 , 2 , . . . , n } . Prelimina- ries SPC product code Kotska Example Numbers Counting Consider t = 6 and n = 3. patterns Conclusions ❳❳❳❳ ✘ P 6 = { (6) , (5 , 1) , (4 , 2) , (4 , 1 , 1) , (3 , 3) , (3 , 2 , 1) , ✘✘✘✘ (3 , 1 , 1 , 1) , (2 , 2 , 2) , ❳ ❳❳❳❳ ✘✘✘✘ (2 , 2 , 1 , 1) , ✭✭✭✭✭ ✘ ❤❤❤❤❤ (2 , 1 , 1 , 1 , 1) , ✭✭✭✭✭✭ ❤❤❤❤❤❤ (1 , 1 , 1 , 1 , 1 , 1) } . ❳ P 3 6 = { (2 , 2 , 2) , (3 , 3 , 0) } S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

  83. Notation SPC product codes under the Definition BEC S. D. Cardell, J. Given two positive integers n and t , the set P n t is a subset of the set of J. Climent, A. López partitions of t of length n defined as Martín P n t = { λ ∈ P t | λ i � = 1 , λ i ≤ n , i = 1 , 2 , . . . , n } . Prelimina- ries SPC product code Kotska Example Numbers Counting Consider t = 6 and n = 3. patterns Conclusions ❳❳❳❳ ✘ P 6 = { (6 , 0 , 0) , (5 , 1 , 0) , (4 , 2 , 0) , (4 , 1 , 1) , (3 , 3 , 0) , (3 , 2 , 1) , ✘✘✘✘ (3 , 1 , 1 , 1) , ❳ (2 , 2 , 2) , ✘✘✘✘ ❳❳❳❳ (2 , 2 , 1 , 1) , ✭✭✭✭✭ ✘ ❤❤❤❤❤ (2 , 1 , 1 , 1 , 1) , ✭✭✭✭✭✭ ❤❤❤❤❤❤ (1 , 1 , 1 , 1 , 1 , 1) } . ❳ P 3 6 = { (2 , 2 , 2) , (3 , 3 , 0) } S. D. Cardell, J. J. Climent, A. López Martín SPC product codes under the BEC

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