New Bounds on the Minimum Distance of Binary LCD Codes of Dimensions 7 to 10
Year: 2026
Type: preprint
Abstract: A binary linear code is linear complementary dual (LCD) if it intersects its dual trivially. We investigate the largest minimum distance of binary LCD codes of dimensions 7 to 10 and lengths 51 to 256. For lower bounds, we develop a local search over column multisets of generator matrices. A column replacement updates all codeword weights in O(2^k) time through the Walsh–Hadamard transform, while the LCD condition is maintained through the Gram matrix. Measured against the published tables closed under the known propagation rules, the search improves the best known lower bounds for 141 parameter pairs. For upper bounds, we first consolidate results from the literature that are not incorporated in the recent tables. We then strengthen the linear program based on the Gauss-sum characterization of LCD codes due to Kang and Xiong by new constraints from the even-weight subcode. Every LP-based nonexistence result is certified in exact rational arithmetic, and the upper bound is lowered for 25 parameter pairs beyond all previously known bounds. The new codes and bounds determine the largest minimum distance for 78 parameter pairs for the first time. All codes were verified by two independent programs, and data and scripts are publicly available.
Author: Promjiraprawat, Kamphol
Institution: Ramkhamhaeng University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225615925
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