On quasi-cyclic codes as a generalization of cyclic codes Morgan - PowerPoint PPT Presentation

On quasi-cyclic codes as a generalization of cyclic codes Morgan Barbier morgan.barbier@unicaen.fr Joint work with: Christophe Chabot Guillaume Quintin University of Caen – GREYC Dinard, C2 October 9th 2012 1 / 23

Outline Generalization of cyclic codes Bijection between QC-codes and some ideals Generator polynomial Q-GRS Definition Decoding Q-BCH Definition Decoding Conclusion 2 / 23

Introduction Definition Let ℓ, m ∈ N . A code C ⊂ F m ℓ is called ℓ -quasi-cyclic of length m ℓ q iff ∀ c = ( c 11 , . . . , c 1 ℓ | . . . | c ( m − 1)1 , . . . , c ( m − 1) ℓ | c m 1 , . . . , c m ℓ ) ∈ C = ⇒ ( c m 1 , . . . , c m ℓ | c 11 , . . . , c 1 ℓ | . . . | c ( m − 1)1 , . . . , c ( m − 1) ℓ ) ∈ C . 3 / 23

Introduction Definition Let ℓ, m ∈ N . A code C ⊂ F m ℓ is called ℓ -quasi-cyclic of length m ℓ q iff ∀ c = ( c 11 , . . . , c 1 ℓ | . . . | c ( m − 1)1 , . . . , c ( m − 1) ℓ | c m 1 , . . . , c m ℓ ) ∈ C = ⇒ ( c m 1 , . . . , c m ℓ | c 11 , . . . , c 1 ℓ | . . . | c ( m − 1)1 , . . . , c ( m − 1) ℓ ) ∈ C . ⇒ C ⊂ ( F q ℓ ) m is cyclic = 3 / 23

Introduction Definition Let ℓ, m ∈ N . A code C ⊂ F m ℓ is called ℓ -quasi-cyclic of length m ℓ q iff ∀ c = ( c 11 , . . . , c 1 ℓ | . . . | c ( m − 1)1 , . . . , c ( m − 1) ℓ | c m 1 , . . . , c m ℓ ) ∈ C = ⇒ ( c m 1 , . . . , c m ℓ | c 11 , . . . , c 1 ℓ | . . . | c ( m − 1)1 , . . . , c ( m − 1) ℓ ) ∈ C . ⇒ C ⊂ ( F q ℓ ) m is cyclic but not necessary F q ℓ -linear. = 3 / 23

Bijection Theorem There is a one-to-one correspondence between ℓ -quasi-cyclic codes over F q of length m ℓ and left ideals of M ℓ ( F q )[ X ] / ( X m − 1) . Sketch of proof : There are one-to-one correspondence between: 1. ℓ -quasi-cyclic codes over F q of length ℓ m 2. submodule of ( F q [ X ] / ( X m − 1)) ℓ 3. left ideal of M ℓ ( F q [ X ] / ( X m − 1)) 4. left ideal of M ℓ ( F q )[ X ] / ( X m − 1). 2 to 3 is given by the Morita equivalence. � 4 / 23

From theory to practice I How to built a ℓ -quasi-cyclic code from a left ideal M ℓ ( F )[ X ] / ( X m − 1)? 5 / 23

From theory to practice I How to built a ℓ -quasi-cyclic code from a left ideal M ℓ ( F )[ X ] / ( X m − 1)? Proposition Let I = � P 1 ( X ) , . . . , P r ( X ) � be a left ideal of M ℓ ( F q )[ X ] / ( X m − 1) . Then the F q -linear space spanned by row k ( X i P j ( X )) : i = 0 , . . . , m − 1 , j = 1 , . . . , r , k = 1 , . . . , ℓ � � is a ℓ -quasi-cyclic code of length m ℓ over F q , where row k : M ℓ ( F q )[ X ] / ( X m − 1) F m ℓ − → q P ( X ) = � m − 1 j =0 P j X j �− → (row k ( P 0 ) , . . . , row k ( P m − 1 )) . 5 / 23

From theory to practice II How to built a left ideal from a ℓ -quasi-cyclic code? 6 / 23

Outline Generalization of cyclic codes Bijection between QC-codes and some ideals Generator polynomial Q-GRS Definition Decoding Q-BCH Definition Decoding Conclusion 7 / 23

Block rank Proposition Let C be an ℓ -quasi-cyclic code over F q of dimension k and length m ℓ . Then there exists an integer r such that 1 ≤ r ≤ k and for all generator matrix G of C and for all i = 0 , . . . , m − 1 , the rank of the i + 1 , . . . , i + ℓ columns of G is r, and is called the block rank . Proposition There exist g 1 , . . . , g r linearly independent vectors of C such that g 1 , . . . , g r , T ( g 1 ) , . . . , T ( g r ) , . . . , T m − 1 ( g 1 ) , . . . , T m − 1 ( g r ) span C. 8 / 23

Generator polynomial Definition (Generator polynomial) Let   g 1 , i ℓ +1 · · · g 1 , ( i +1) ℓ . . . .   . . G i =  ∈ M ℓ ( F q ) ,     · · · g r , i ℓ +1 g r , ( i +1) ℓ  0 and ν the smallest integer such that G ν � = 0. We call m − 1 g ( X ) = 1 G i X i ∈ M ℓ ( F q )[ X ] , � X ν i =0 the generator polynomial of C and C corresponds to the left ideal spanned by � g ( X ) � . 9 / 23

Property Proposition Let C be an ℓ -quasi-cyclic code of length m ℓ over F q . Let P ( X ) be a generator polynomial of C and Q ( X ) a generator polynomial of its dual. Then mod ( X m − 1) P ( X ) t Q ∗ ( X ) ≡ 0 where Q ∗ ( X ) = X deg Q Q (1 / X ) denotes the reciprocal polynomial of Q and t Q the polynomial whose coefficients are the transposed matrices of the coefficients of Q. 10 / 23

Outline Generalization of cyclic codes Bijection between QC-codes and some ideals Generator polynomial Q-GRS Definition Decoding Q-BCH Definition Decoding Conclusion 11 / 23

Definition (Generalized Reed-Solomon codes) Let R be a finite ring, n ≥ k ∈ N be two integers, ( x i ) i =1 ,..., n ∈ R n be a subtractive set, and v i ∈ R n be n invertible elements of R . We define GRS ll ( v , x , k ) = { ( v 1 ev l ( P , x 1 ) , . . . , v n ev l ( P , x n )) : P ∈ R [ X ] < k } . We can also define 3 other GRS codes. 12 / 23

Outline Generalization of cyclic codes Bijection between QC-codes and some ideals Generator polynomial Q-GRS Definition Decoding Q-BCH Definition Decoding Conclusion 13 / 23

From theory... Remark (Thanks to Coste) Let R be any finite ring, n < m be two positive integers and M ∈ M n × m ( R ) . Then there exists a nonzero x ∈ R m such that Mx = 0 . Proposition Let P ∈ R [ X ] of degree at most n with at least n + 1 roots contained in a commutative subtractive subset of A. Then P = 0 . 14 / 23

...to practice Algorithm 1: Welch-Berlekamp � n − k : A received vector y of R n with at most t = � Input errors. 2 Output : The unique codeword within distance t of y . d´ ebut y ′ ← ( v − 1 1 y 1 , . . . , v − 1 n y n ), compute Q = Q 0 ( X ) + Q 1 ( X ) Y ∈ ( R [ X ])[ Y ] 1. Q ( x i , y ′ i ) = 0 for all 1 ≤ i ≤ n , 2. deg Q 0 ≤ n − t − 1, 3. deg Q 1 ≤ n − t − 1 − ( k − 1). 4. The leading coefficient of Q 1 is 1 A . P ← the unique root of Q in R [ X ] < k , return ( v 1 ev l ( P , x 1 ) , . . . , v n ev l ( P , x n )) . 15 / 23

Outline Generalization of cyclic codes Bijection between QC-codes and some ideals Generator polynomial Q-GRS Definition Decoding Q-BCH Definition Decoding Conclusion 16 / 23

A primitive root of unity Definition Let q be a prime power. A matrix A ∈ M ℓ ( F q s ) is called a primitive m-th root of unity if ◮ A m = I ℓ , ◮ A i � = I ℓ if i < m , ◮ det( A i − A j ) � = 0, whenever i � = j , that is power of A are a subtractive set. 17 / 23

Quasi-BCH codes Definition (Left quasi-BCH codes) Let A be a primitive m -th root of unity in M ℓ ( F q s ) and δ ≤ m . We define the left ℓ -quasi-BCH code of length m ℓ , with respect to A , with designed minimum distance δ , over F q by Q-BCH l ( m , ℓ, δ, A ) =   m − 1  q ) m :  �  ( c 1 , . . . , c m ) ∈ ( F ℓ A ij c j +1 = 0 for i = 1 , . . . , δ − 1  . j =0 Similarly, we can define the right ℓ -quasi-BCH codes . 18 / 23

Outline Generalization of cyclic codes Bijection between QC-codes and some ideals Generator polynomial Q-GRS Definition Decoding Q-BCH Definition Decoding Conclusion 19 / 23

Q-BCH code as a cyclic RS code Proposition A study of the orthogonal codes gives row 1 (RS l (( A i ) i =1 ,..., m , m − δ + 1)) Q-BCH l ( m , ℓ, δ, A ) = row 1 (RS r (( A i ) i =1 ,..., m , m − δ + 1)) . Q-BCH r ( m , ℓ, δ, A ) = = ⇒ Use the Welch-Berlekamp algorithm to decode the Q-BCH codes. 20 / 23

Conclusion I New codes over F 4 [171 , 11 , 109] F 4 [172 , 11 , 110] F 4 [173 , 11 , 110] F 4 [174 , 11 , 111] F 4 [175 , 11 , 112] F 4 [176 , 11 , 113] F 4 [177 , 11 , 114] F 4 [178 , 11 , 115] F 4 [179 , 11 , 115] F 4 [180 , 11 , 116] F 4 [181 , 11 , 117] F 4 [182 , 11 , 118] F 4 [183 , 11 , 119] F 4 [184 , 10 , 121] F 4 [184 , 11 , 120] F 4 [185 , 10 , 122] F 4 [185 , 11 , 121] F 4 [186 , 10 , 123] F 4 [186 , 11 , 122] F 4 [187 , 10 , 124] F 4 [187 , 11 , 123] F 4 [188 , 10 , 125] F 4 [188 , 11 , 124] F 4 [189 , 10 , 126] F 4 [189 , 11 , 125] F 4 [190 , 10 , 127] F 4 [190 , 11 , 126] F 4 [191 , 10 , 128] F 4 [191 , 11 , 127] F 4 [192 , 11 , 128] F 4 [193 , 11 , 128] F 4 [194 , 11 , 128] F 4 [195 , 11 , 128] F 4 [196 , 11 , 129] F 4 [197 , 11 , 130] F 4 [198 , 11 , 130] F 4 [199 , 11 , 131] F 4 [200 , 11 , 132] F 4 [201 , 10 , 133] F 4 [201 , 11 , 132] F 4 [202 , 10 , 134] F 4 [202 , 11 , 132] F 4 [203 , 10 , 135] F 4 [204 , 10 , 136] F 4 [204 , 11 , 133] F 4 [205 , 11 , 134] F 4 [210 , 11 , 137] F 4 [213 , 11 , 139] F 4 [214 , 11 , 140] F 4 Table: 49 new codes over F 4 which have a larger minimum distance than the previously known ones. 21 / 23

Conclusion II ◮ 49 new best codes. ◮ Unique and list decoding algorithms faster on valuation rings (e.g. Galois rings) than finite fields. ◮ Generalization of well known results on cyclic codes over finite fields for cyclic codes over finite rings, with application to quasi-cyclic codes: ◮ Correspondence between QC codes and some ideals. ◮ Generator polynomials. ◮ Two new classes of codes with decoding algorithm. ◮ Orthogonality of these classes of codes. ◮ Weight enumerator distribution. 22 / 23

On quasi-cyclic codes as a generalization of cyclic codes Morgan Barbier morgan.barbier@unicaen.fr Joint work with: Christophe Chabot Guillaume Quintin University of Caen – GREYC Dinard, C2 October 9th 2012 23 / 23