Jon-Lark Kim

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

Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions

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Higher-Order CIS Codes

. 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.

Some bounds on binary LCD codes

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function response, and extend classical coding methods to construct multiply constant-weight codes from known (q) -ary and constant-weight codes. We derive analogs of Johnson bounds and give constructions showing these bounds to be asymptotically tight up to a constant factor under certain conditions. We also examine the rates of multiply constant-weight codes and demonstrate that these rates are the same as those of constant-weight codes of corresponding parameters.

Multiply constant weight codes

We introduce complementary information set codes of higher order. A binary linear code of length tk and dimension k is called a complementary information set code of order t (t-CIS code for short) if it has t pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a t-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order 3 is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length ≤ 12 is given. Nonlinear codes better than linear codes are derived by taking binary images of Z4-codes. A general algorithm based on Edmonds basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate 1/t, it either provides t disjoint information sets or proves that the code is not t-CIS. Using this algorithm, all optimal or best known [tk, k] codes, where t = 3, 4, . . . , 256 and 1≤ k ≤⌊256/t⌋ are shown to be t-CIS for all such k and t, except for t = 3 with k = 44 and t = 4 with k = 37.

Classification of Binary Self-Dual (48,24,10) Codes with an Automorphism of Odd Prime Order

A linear code with a complementary dual (or An LCD code) is defined to be a linear code C whose dual code C⊥ satisfies C ∩ C⊥= 0

Codes over rings and Hermitian lattices

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Optimal subcodes and optimum distance profiles of self-dual codes

. Let LD (n, k) denote the maximum of possible values of d among [n, k, d] binary LCD codes. We give the exact values of LD (n, k) for k = 2 for all n and some bounds on LD (n, k) for other cases. From our results and some direct search we obtain a complete table for the exact values of LD (n, k) for 1 ≤ k ≤ n ≤ 12. As a consequence, we also derive bounds on the dimensions of LCD codes with fixed lengths and minimum distances.

Optimal subcodes of formally self-dual codes and their optimum distance profiles

The function M(m, n, d, w), the largest size of an unrestricted binary code made of m by n arrays, with constant row weight w, and minimum distance d is introduced and compared to the classical functions of combinatorial coding theory Aq(n, d) and A(n, d, w). The analogues for systematic codes of A(n, d) and A(n, d, w) are introduced apparently for the first time. An application to the security of embedded systems is given: these codes happen to be efficient challenges for physically unclonable functions.

A New LRPC-Kronecker Product Codes Based Public-Key Cryptography

The purpose of this paper is to complete the classification of binary self-dual [48,24,10] codes with an automorphism of odd prime order. We prove that if there is a self-dual [48, 24,10] code with an automorphism of type p-(c,f) with p being an odd prime, then p = 3,c = 16,f = 0. By considering only an automorphism of type 3-(16,0), we prove that there are exactly 264 inequivalent self-dual [48,24,10] codes with an automorphism of odd prime order, equivalently, there are exactly 264 inequivalent cubic self-dual [48,24,10] codes.