Lin Sok

The combinatorics of LCD codes: linear programming bound and orthogonal matrices

Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings

Classification of Extremal and

R_k

s

. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.

-Extremal Binary Self-Dual Codes of Length 38

In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal self-dual codes, two s-extremal codes, and 1730 s-extremal codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of -extremal codes permits to achieve the classification of all -extremal codes with .

Towards the classification of self-dual bent functions in eight variables

In this paper, we classify quadratic and cubic self-dual bent functions in eight variables with the help of computers. There are exactly four and 45 non-equivalent self-dual bent functions of degree two and three, respectively. This result is achieved by enumerating all eigenvectors with ± 1 entries of the Sylvester Hadamard matrix with an integer programming algorithm based on lattice basis reduction. The search space has been reduced by breaking the symmetry of the problem with the help of additional constraints. The final number of non-isomorphic self-dual bent functions has been determined by exploiting that EA-equivalence of Boolean functions is related to the equivalence of linear codes.

Lattice based codes for insertion and deletion channels

Insertion/Deletion codes for the Levenshtein distance are constructed by truncation of lattices for the L1 metric. These lattices are obtained from Construction A applied to binary codes and Z4-codes. Finally, Gilbert and Hamming type of bounds are derived.

On formally self-dual boolean functions in 2,4 and 6 variables

In this paper, we classify all formally self-dual Boolean functions and self-dual bent functions under the action of the extended symmetric group in 2,4 variables, and give a lower bound for the number of non-equivalent functions in 6 variables. There are exactly 2,91 (1,3 respectively) and at least 5535376 representatives from equivalence class of formally self-dual Boolean functions (self-dual bent functions respectively).

Self-dual codes and orthogonal matrices over large finite fields

Abstract In this paper, we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, code extension and projection over a self-dual basis are the main tools. Some optimal, almost MDS, and MDS self-dual codes over both small and large finite fields are constructed. Moreover, over fifty MDS codes with new parameters are constructed. Comparisons with classical constructions are made.

On constacyclic codes over

Abstract We first define a new Gray map from R = Z 4 + u Z 4 to Z 4 2 , where u 2 = 1 and study ( 1 + 2 u ) -constacyclic codes over R. Also of interest are some properties of ( 1 + 2 u ) -constacyclic codes over R. Considering their Z 4 images, we prove that the Gray images of ( 1 + 2 u ) -constacyclic codes of length n over R are cyclic codes of length 2n over Z 4 . In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic R-codes published by Ozen et al. [7] .

\mathbb{Z}_4[u]/\langle u^2-1\rangle

The Gilbert type bound for codes in the title is reviewed, both for small and large alphabets. Constructive lower bounds better than these existential bounds are derived from geometric codes, either over