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.        

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.

Linear codes from support designs of ternary cyclic codes

2022 ◽  
Author(s):  
Pan Tan ◽  
Cuiling Fan ◽  
Sihem Mesnager ◽  
Wei Guo
Keyword(s):  
Linear Codes ◽  
Cyclic Codes

scholarly journals Projection Decoding of Some Binary Optimal Linear Codes of Lengths 36 and 40

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 15
Author(s):  
Lucky Galvez ◽  
Jon-Lark Kim
Keyword(s):  
Linear Codes ◽  
Cyclic Codes ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Decoding Algorithm ◽  
Decoding Algorithms ◽  
Syndrome Decoding ◽  
Optimal Linear ◽  
Optimal Linear Codes ◽  
Efficient Decoding

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F ( 4 ) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical.

A note on constant weight cyclic codes

1973 ◽  
Vol 18 (1) ◽  
pp. 217-220
Author(s):  
HARRY T. HSU ◽  
JOHN E. OLSON
Keyword(s):  
Cyclic Codes ◽  
Constant Weight

DNA computing over the ring ℤ4[v]/〈v2 − v〉

2018 ◽  
Vol 11 (07) ◽  
pp. 1850090
Author(s):  
Narendra Kumar ◽  
Abhay Kumar Singh
Keyword(s):  
Significant Role ◽  
Dna Computing ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Finite Ring ◽  
Dna Codes ◽  
Reversible Cyclic Codes ◽  
Text Images

In this paper, we discuss the DNA construction of general length over the finite ring [Formula: see text], with [Formula: see text], which plays a very significant role in DNA computing. We discuss the GC weight of DNA codes over [Formula: see text]. Several examples of reversible cyclic codes over [Formula: see text] are provided, whose [Formula: see text]-images are [Formula: see text]-linear codes with good parameters.

Linear codes from vectorial Boolean functions in the context of algebraic attacks

2020 ◽  
pp. 2150032
Author(s):  
M. Boumezbeur ◽  
S. Mesnager ◽  
K. Guenda
Keyword(s):  
Boolean Functions ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Weight Enumerator ◽  
Direct Link ◽  
Algebraic Attacks ◽  
Vectorial Boolean Functions ◽  
Lcd Codes ◽  
Vectorial Function ◽  
The Relationship

In this paper, we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain [Formula: see text]-ary cyclic codes (which we show that they are LCD codes). We also present some properties of those cyclic codes as well as their weight enumerator. In addition, we generalize the so-called algebraic complement and study its properties.

On cyclic codes in incidence rings

2006 ◽  
Vol 43 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Ricardo Alfaro ◽  
Andrei V. Kelarev
Keyword(s):  
Directed Graph ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Matrix Ring ◽  
Hamming Weight ◽  
Quotient Rings

Cyclic codes are defined as ideals in polynomial quotient rings. We are using a matrix ring construction in a similar way to define classes of codes. It is shown that all cyclic and all linear codes can be embedded as ideals in this construction. A formula for the largest Hamming weight of one-sided ideals in incidence rings is given. It is shown that every incidence ring defined by a directed graph always possesses a principal one-sided ideal that achieves the optimum Hamming weight.

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.

New linear codes derived from binary cyclic codes of length 151

2006 ◽  
Vol 153 (5) ◽  
pp. 581 ◽  
Author(s):  
C. Tjhai ◽  
M. Tomlinson ◽  
M. Grassl ◽  
R. Horan ◽  
M. Ahmed ◽  
...  
Keyword(s):  
Linear Codes ◽  
Cyclic Codes
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