A Study of Burst-Error-Correcting Array Codes D´ ebora Beatriz Claro Zanitti Undergraduate Student of Telecommunication Engineering at S˜ ao Paulo State University - UNESP Scholarship: FAPESP 2017/17948-8
◮ Poster proposal: The purpose of this presentation is to study the encoding and decoding of array codes to correct burst errors. These codes have application in telecommunication area, for example: magnetic recording, storage system, audio compression, image processing, etc.
◮ Array Codes: Array codes are two-dimensional codes, in which the parity check equations are given by exclusive-OR operations over lines in one or more direction, reducing the overall complexity and allowing fast encoding and decoding. They are linear codes that can be either convolutional or block-type. Block array codes A ( n 1 , n 2 ) consist of all n 1 × n 2 matrices in which all row-sums and all column-sums (or certain diagonal-sums) have even parity. The dimensional of this codes in ( n 1 − 1) × ( n 2 − 1) and they have minimum Hamming distance 4. ◮ Burst Errors Array codes interest lies in their ability to correct clusters of errors called bursts, which are errors that occur in many consecutive bits rather than occurring in bits independently of each other.
◮ Encoding: Let A (5 , 5) be a block array code. Consider the vector u = 10100101101110111111 ∈ A (5 , 5). First, we put this vector row-wise into a (5 × 4) array and we added a fifth column formed by the diagonal syndrome. Figura: Encoding. The diagonal syndrome is thus obtained: ◮ 5 is the parity digit for 9, 13, 17, and 21; ◮ 10 is the parity digit for 1, 14, 18, and 22; ◮ 15 is the parity digit for 2, 6, 19, and 23; ◮ 20 is the parity digit for 3, 7, 11, and 24; ◮ 25 is the parity digit for 4, 8, 12, and 16.
◮ Decoding: Now, suppose we received the following array in A (5 , 5): Figura: Decoding. The array has a burst error on the first line. First we find the vertical and horizontal syndromes through by exclusive-OR operations. The vertical syndrome is S v = 11110, while its diagonal syndrome is S d = 11110 (read back). As the two syndromes are equal, we should not do any shift, so to correct it we must add S v the first line of the code. 01011 + s v = 01011 + 11110 = 10101(mod 2).
THANK YOU! Acknowledgements
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