The structure of ℤ2ℤ2s-additive cyclic codes

2018 ◽  
Vol 10 (04) ◽  
pp. 1850048 ◽  
Author(s):  
Ismail Aydogdu ◽  
Taher Abualrub
Keyword(s):  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Optimal Parameter ◽  
Binary Codes ◽  
Binary Linear Codes ◽  
Additive Codes

[Formula: see text]-additive codes for any integer [Formula: see text] are considered as codes over mixed alphabets. They are a generalization of binary linear codes and linear codes over [Formula: see text] In this paper, we are interested in studying [Formula: see text]-additive cyclic codes. We will give the generator polynomials of these codes. We will also give the minimal spanning sets for these codes. We will define separable [Formula: see text]-additive codes and provide conditions on the generator polynomials for a [Formula: see text]-additive cyclic code to be separable. Finally, we present some examples of optimal parameter binary codes obtained as images of [Formula: see text]-additive cyclic codes.

New extremal Type II ℤ4-codes of length 32 obtained from Hadamard matrices

2019 ◽  
Vol 11 (05) ◽  
pp. 1950057
Author(s):  
Sara Ban ◽  
Dean Crnković ◽  
Matteo Mravić ◽  
Sanja Rukavina
Keyword(s):  
Incidence Matrix ◽  
Linear Codes ◽  
Minimum Weight ◽  
Hadamard Matrices ◽  
Binary Codes ◽  
Type Ii ◽  
Hadamard Design ◽  
Binary Linear Codes ◽  
Type Formula ◽  
Hadamard Designs

For every Hadamard design with parameters [Formula: see text]-[Formula: see text] having a skew-symmetric incidence matrix we give a construction of 54 Hadamard designs with parameters [Formula: see text]-[Formula: see text]. Moreover, for the case [Formula: see text] we construct doubly-even self-orthogonal binary linear codes from the corresponding Hadamard matrices of order 32. From these binary codes we construct five new extremal Type II [Formula: see text]-codes of length 32. The constructed codes are the first examples of extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text], [Formula: see text], whose residue codes have minimum weight 8. Further, correcting the results from the literature we construct 5147 extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text].

scholarly journals Enumeration of self-dual cyclic codes of some specific lengths over finite fields

2018 ◽  
Vol 10 (03) ◽  
pp. 1850031 ◽  
Author(s):  
Supawadee Prugsapitak ◽  
Somphong Jitman
Keyword(s):  
Characteristic Function ◽  
Finite Field ◽  
Finite Fields ◽  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Efficient Algorithms ◽  
Important Class ◽  
Field Characteristic ◽  
Positive Integers

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length [Formula: see text] over a finite field if and only if [Formula: see text] and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for self-dual cyclic codes of length [Formula: see text] over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are positive integers. Some number theoretical tools are established. Based on these results, alternative formulas and efficient algorithms to determine the number of self-dual cyclic codes of such lengths are provided.

Triple Cyclic Codes Over 𝔽q + u𝔽q

2021 ◽  
pp. 1-21
Author(s):  
Ting Yao ◽  
Shixin Zhu ◽  
Binbin Pang
Keyword(s):  
Special Class ◽  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Cyclic Shift ◽  
Generating Sets ◽  
Optimal Linear ◽  
Number Formula ◽  
The Relationship ◽  
Optimal Linear Codes

Let [Formula: see text], where [Formula: see text] is a power of a prime number [Formula: see text] and [Formula: see text]. A triple cyclic code of length [Formula: see text] over [Formula: see text] is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as [Formula: see text]-submodules of [Formula: see text]. In this paper, we study the generator polynomials and the minimum generating sets of this kind of codes. Some optimal or almost optimal linear codes are obtained from this family of codes. We present the relationship between the generators of triple cyclic codes and their duals. As a special class of triple cyclic codes, separable codes over [Formula: see text] are discussed briefly in the end.

Some results on the linear codes over the finite ring F2+v1F2+⋯+vrF2

2016 ◽  
Vol 14 (01) ◽  
pp. 1650012 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Finite Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Quantum Error ◽  
Quasi Cyclic Codes

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring [Formula: see text], [Formula: see text] for [Formula: see text]. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over [Formula: see text] are obtained. A necessary and sufficient condition for cyclic code over [Formula: see text] that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over [Formula: see text].

scholarly journals The structure of one weight linear and cyclic codes over Z_{2}^r x (Z_{2} + uZ_{2})^s

2017 ◽  
Vol 8 (1) ◽  
pp. 92-101 ◽  
Author(s):  
Ismail Aydogdu
Keyword(s):  
Linear Codes ◽  
Cyclic Codes ◽  
Constant Weight ◽  
Additive Codes ◽  
Weight One

Inspired by the Z2Z4-additive codes, linear codes over Z_{2}^r x  (Z_{2} + uZ_{2})^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z_{2}^r x  (Z_{2} + uZ_{2})^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.        

QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER $\mathbb{F}_4 + u\mathbb{F}_4$

2011 ◽  
Vol 09 (02) ◽  
pp. 689-700 ◽  
Author(s):  
XIAOSHAN KAI ◽  
SHIXIN ZHU
Keyword(s):  
Cyclic Code ◽  
Cyclic Codes ◽  
Binary Codes ◽  
Quantum Codes ◽  
Quantum Code ◽  
Orthogonal Codes

We give a construction for quantum codes from linear and cyclic codes over [Formula: see text]. We derive Hermitian self-orthogonal codes over [Formula: see text] as Gray images of linear and cyclic codes over [Formula: see text]. In particular, we use two binary codes associated with a cyclic code over [Formula: see text] of odd length to determine the parameters of the corresponding quantum code.

scholarly journals Simple Tight Performance Bound Based on the GFBT for Binary Linear Codes

2020 ◽  
Vol 428 ◽  
pp. 012028
Author(s):  
Jia Liu ◽  
Mingyu Zhang ◽  
Rongjun Chen ◽  
Chaoyong Wang ◽  
Xiaofeng An ◽  
...  
Keyword(s):  
Linear Codes ◽  
Performance Bound ◽  
Binary Linear Codes
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