Polyadic Constacyclic Codes Yun Fan Department of Mathematics, - PowerPoint PPT Presentation

Polyadic Constacyclic Codes Yun Fan Department of Mathematics, Central China Normal University, Wuhan, 430079, CHINA Email: yfan@mail.ccnu.edu.cn Tel: (+86) 15002714631 A joint work with Bocong Chen, Hai Q. Dinh, San Ling 1 / 38

Abstract Necessary and sufficient conditions of the existence of polyadic constacyclic codes are reported in terms of p -valuations; the main ideas to solve the question are described. Some consequences are derived, and some examples are exhibited. 2 / 38

Contents Linear codes, Cyclic Codes 1 Constacyclic Codes 2 Polyadic Constacyclic Codes 3 Main Results 4 Applications 5 3 / 38

Linear codes, Cyclic Codes Hamming Weight, Hamming Distance F q : finite field with q elements, q a prime power F ∗ q : the multiplicative group of nonzero elements of F q � � � � a i ∈ F q F n q : = ( a 0 , a 1 , ··· , a n − 1 ) , the string a = ( a 0 , a 1 , ··· , a n − 1 ) is called a word over the alphabet F q 4 / 38

Linear codes, Cyclic Codes Hamming Weight, Hamming Distance F q : finite field with q elements, q a prime power F ∗ q : the multiplicative group of nonzero elements of F q � � � � a i ∈ F q F n q : = ( a 0 , a 1 , ··· , a n − 1 ) , the string a = ( a 0 , a 1 , ··· , a n − 1 ) is called a word over the alphabet F q Definition Hamming weight : w ( a ) := # { i | 0 ≤ i < n , a i � = 0 } , ∀ a = ( a 0 , a 1 , ··· , a n − 1 ) ∈ F n q Hamming distance : d ( a , a ′ ) := w ( a − a ′ ) , ∀ a , a ′ ∈ F n q 4 / 38

Linear Codes Definition Any given subspace C of F n q is said to be a linear code of length n over the alphabet F q ; any c = ( c 0 , c 1 , ··· , c n − 1 ) ∈ C is said to be a codeword . 5 / 38

Linear Codes Definition Any given subspace C of F n q is said to be a linear code of length n over the alphabet F q ; any c = ( c 0 , c 1 , ··· , c n − 1 ) ∈ C is said to be a codeword . k := dim C , dimension of the linear code c � = c ′ ∈ C d ( c , c ′ ), d = d ( C ) := min the minimum distance of the linear code w ( C ) := min 0 � = c ∈ C w ( c ), the minimum weight of the linear code C is said to be an [ n , k , d ] code. 5 / 38

Linear Codes Definition Any given subspace C of F n q is said to be a linear code of length n over the alphabet F q ; any c = ( c 0 , c 1 , ··· , c n − 1 ) ∈ C is said to be a codeword . k := dim C , dimension of the linear code c � = c ′ ∈ C d ( c , c ′ ), d = d ( C ) := min the minimum distance of the linear code w ( C ) := min 0 � = c ∈ C w ( c ), the minimum weight of the linear code C is said to be an [ n , k , d ] code. Remark. For linear codes, d ( C ) = w ( C ). 5 / 38

What Codes Are Good Codes Both the dimension and the minimum distance of the code are bigger. However, the dimension and the minimum distance of a code restrict each other. So the question is: how to trade off them. 6 / 38

What Codes Are Good Codes Both the dimension and the minimum distance of the code are bigger. However, the dimension and the minimum distance of a code restrict each other. So the question is: how to trade off them. The code has good mathematical structure 6 / 38

Orthogonality, Self-duality Euclidean inner product: n − 1 ∀ a = ( a 0 , ··· , a n − 1 ) , a ′ = ( a ′ � a , a ′ � := a i a ′ 0 , ··· , a ′ n − 1 ) ∈ F n ∑ i , q i =0 Definition C ≤ F n q 7 / 38

Orthogonality, Self-duality Euclidean inner product: n − 1 ∀ a = ( a 0 , ··· , a n − 1 ) , a ′ = ( a ′ � a , a ′ � := a i a ′ 0 , ··· , a ′ n − 1 ) ∈ F n ∑ i , q i =0 Definition C ≤ F n q � � � C ⊥ := � � a , c � = 0 , ∀ c ∈ C a ∈ F n , is called the orthogonal q code of C . 7 / 38

Orthogonality, Self-duality Euclidean inner product: n − 1 ∀ a = ( a 0 , ··· , a n − 1 ) , a ′ = ( a ′ � a , a ′ � := a i a ′ 0 , ··· , a ′ n − 1 ) ∈ F n ∑ i , q i =0 Definition C ≤ F n q � � � C ⊥ := � � a , c � = 0 , ∀ c ∈ C a ∈ F n , is called the orthogonal q code of C . C is said to be self-orthogonal if C ⊆ C ⊥ . 7 / 38

Orthogonality, Self-duality Euclidean inner product: n − 1 ∀ a = ( a 0 , ··· , a n − 1 ) , a ′ = ( a ′ � a , a ′ � := a i a ′ 0 , ··· , a ′ n − 1 ) ∈ F n ∑ i , q i =0 Definition C ≤ F n q � � � C ⊥ := � � a , c � = 0 , ∀ c ∈ C a ∈ F n , is called the orthogonal q code of C . C is said to be self-orthogonal if C ⊆ C ⊥ . C is said to be self-dual if C = C ⊥ . 7 / 38

Cyclic Codes, Constacyclic Codes C ≤ F n q : a linear code Definition A linear code C is called a cyclic code if ( c n − 1 , c 0 , ··· , c n − 2 ) ∈ C , ∀ ( c 0 , c 1 , ··· , c n − 1 ) ∈ C 8 / 38

Cyclic Codes, Constacyclic Codes C ≤ F n q : a linear code Definition A linear code C is called a cyclic code if ( c n − 1 , c 0 , ··· , c n − 2 ) ∈ C , ∀ ( c 0 , c 1 , ··· , c n − 1 ) ∈ C λ ∈ F ∗ q Definition A linear code C is called a λ -constacyclic code if ( λ c n − 1 , c 0 , ··· , c n − 2 ) ∈ C , ∀ ( c 0 , c 1 , ··· , c n − 1 ) ∈ C λ = 1: λ -constacyclic code = cyclic code λ = − 1: λ -constacyclic code is called negacyclic code 8 / 38

Huge literatures on cyclic codes: BCH codes RS codes Berlekamp Algorithm ··· 9 / 38

Huge literatures on cyclic codes: BCH codes RS codes Berlekamp Algorithm ··· Quadratic residue codes Duadic codes Polyadic codes ··· 9 / 38

Constacyclic Codes How to Determine Constacyclic Codes λ ∈ F ∗ q F q [ X ]: the polynomial algebra over F q � X n − λ � : the ideal of F q [ X ] generated by X n − λ F q [ X ] / � X n − λ � : the quotient algebra F q [ X ] / � X n − λ � linear iso . F n − → q a 0 + a 1 X + ··· + a n − 1 X n − 1 �− → ( a 0 , a 1 , ··· , a n − 1 ) 10 / 38

Constacyclic Codes How to Determine Constacyclic Codes λ ∈ F ∗ q F q [ X ]: the polynomial algebra over F q � X n − λ � : the ideal of F q [ X ] generated by X n − λ F q [ X ] / � X n − λ � : the quotient algebra F q [ X ] / � X n − λ � linear iso . F n − → q a 0 + a 1 X + ··· + a n − 1 X n − 1 �− → ( a 0 , a 1 , ··· , a n − 1 ) C ≤ F n q is a λ -constacyclic code if and only if the following set � c 0 + c 1 X + ··· + c n − 1 X n − 1 � � � ( c 0 , c 1 , ··· , c n − 1 ) ∈ C is an ideal of F q [ X ] / � X n − λ � . 10 / 38

Since F q [ X ] is a principal ideal ring, one gets that Proposition λ -constacyclic codes C over F q of length n are one to one corresponding to the monomial divisors g ( X ) of X n − λ such that � � � � f ( X ) g ( X ) ( mod X n − λ ) C = � f ( X ) ∈ F q [ X ] 11 / 38

Since F q [ X ] is a principal ideal ring, one gets that Proposition λ -constacyclic codes C over F q of length n are one to one corresponding to the monomial divisors g ( X ) of X n − λ such that � � � � f ( X ) g ( X ) ( mod X n − λ ) C = � f ( X ) ∈ F q [ X ] Definition Let X n − λ = g ( X ) h ( X ) and � � � � f ( X ) g ( X ) ( mod X n − λ ) C = � f ( X ) ∈ F q [ X ] . Then g ( X ) is called a generator polynomial of C , while h ( X ) is called a check polynomial of C . � � � � c ( X ) ∈ F q [ X ] / � X n − λ � � c ( X ) h ( X ) ≡ 0 ( mod X n − λ ) C = . 11 / 38

How to Determine the Divisors of X n − λ From now on we always assume that gcd( q , n ) = 1 12 / 38

How to Determine the Divisors of X n − λ From now on we always assume that gcd( q , n ) = 1 Z n : the residue ring of the integer ring Z modulo n Z ∗ n : the multiplicative group of units of Z n 12 / 38

How to Determine the Divisors of X n − λ From now on we always assume that gcd( q , n ) = 1 Z n : the residue ring of the integer ring Z modulo n Z ∗ n : the multiplicative group of units of Z n q ( λ ), the order of λ in the group F ∗ r := ord F ∗ q , hence r | ( q − 1) and q ∈ Z ∗ nr nr , hence nr | ( q e − 1) nr ( q ), the order of q in the group Z ∗ e := ord Z ∗ ω ∈ F q e : a primitive nr -th root of unity 12 / 38

How to Determine the Divisors of X n − λ From now on we always assume that gcd( q , n ) = 1 Z n : the residue ring of the integer ring Z modulo n Z ∗ n : the multiplicative group of units of Z n q ( λ ), the order of λ in the group F ∗ r := ord F ∗ q , hence r | ( q − 1) and q ∈ Z ∗ nr nr , hence nr | ( q e − 1) nr ( q ), the order of q in the group Z ∗ e := ord Z ∗ ω ∈ F q e : a primitive nr -th root of unity 1+ r Z nr = { 1+ rk | k = 0 , 1 , ··· , n − 1 } ⊆ Z nr X n − λ = ∏ ( X − ω i ) i ∈ 1+ r Z nr Proof. ( ω i ) n = λ if and only if i ≡ 1 ( mod r ). 12 / 38

X n − λ = ( X − ω i ) is not a decomposition in F q [ X ] in ∏ i ∈ 1+ r Z nr general ! 13 / 38

X n − λ = ( X − ω i ) is not a decomposition in F q [ X ] in ∏ i ∈ 1+ r Z nr general ! A polynomial f ( X ) ∈ F q e [ X ] belongs to F q [ X ] if and only if it is invariant by the Galois group Gal ( F q e / F q ), which is a cyclic group generated by → α q F q e − γ q : → F q e , α �− 13 / 38