Optimal Binary Linear Complementary Pairs of Codes

Choi, Whan-Hyuk
Kim, Jon-Lark
Özbudak, Ferruh
A pair of linear codes (C, D) of length n over F-q is called a linear complementary pair (LCP) if their direct sum yields the full space F-q(n). By a result of Carlet et al. (2019), the best security parameters of binary LCPs of codes are left open. Motivated by this, we study binary LCPs of codes. We describe a sufficient condition for binary LCPs of codes which are not optimal. We carry out an exhaustive search to determine the best security parameters for binary LCPs of codes up to length 18. We also obtain results on optimal binary LCPs of codes for infinitely many parameters. For any k >= 2 and length n congruent to 0 or 1 mod (2(k) - 1), we prove that binary [n, k] LCPs of codes are optimal. Binary LCPs of codes of dimensions 2, 3, and 4 are also optimal for all lengths except for two instances, when (n, k) = (4, 3) and (8, 4). We provide explicit constructions of these infinite families of optimal LCPs. Our results also indicate that many security parameters coming from binary LCPs of codes exceed those from binary LCD codes by 1 or 2.


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LCD codes from tridiagonal Toeplitz matrices
Shi, Minjia; Özbudak, Ferruh; Xu, Li; Solé, Patrick (2021-10-01)
Double Toeplitz (DT) codes are codes with a generator matrix of the form (I,T) with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields.
Citation Formats
W.-H. Choi, C. GÜNERİ, J.-L. Kim, and F. Özbudak, “Optimal Binary Linear Complementary Pairs of Codes,” CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES, pp. 0–0, 2022, Accessed: 00, 2023. [Online]. Available: https://hdl.handle.net/11511/101740.