For $q\in\set{2,3}$, we show that a $k$-dimensional linear code over the finite field $\F_q$ of order $q$ is linear complementary dual (LCD) exactly when one root-of-unity value of its weight enumerator has magnitude $q^{k/2}$. We convert the phase of this value, together with the parity type in the binary case, into exact linear constraints on the weight distribution and incorporate them into a Gauss-phase linear program. The resulting program uses only the ordinary weight distributions of the code and its dual and adds only a constant-size set of branch equations to the usual Hamming/MacWilliams constraints, so it remains close in size to the standard Hamming LP while retaining additional arithmetic information. Computations over the audited binary and ternary ranges show systematic strengthening of the Hamming LCD relaxation. In the binary case, comparison with the established mixed joint-weight-enumerator LP yields four strict improvements, lowering the benchmark upper bound by one in each case. Each strict comparison is verified exactly by rational feasibility witnesses and integer Farkas certificates.
The generalized packing--covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. W...
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 2 citations· ⚡1
Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whene...
Javier de la Cruz, A. Horlemann, Marc Newman et al.· 0 citations
We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most $(J+1)!$, where $J$ is the column weight. For every fixed pair of column and row weights $2\le J<L$, we show that thi...
In this paper, we derive complete weight distributions and solve constrained optimization problems for affine-half coordinate extensions of a known family of partial-spread minimal binary codes. The extended code has length 2m+2m−1−1 and dimension m+1, and it remains minimal. The repetition step and its preservation of...
A storage code on a graph assigns a symbol to each vertex so that the symbol can be recovered from its neighbors. We prove that the binary full-parity storage codes on triangle-free coset graphs of primitive BCH codes have rate tending to one for every fixed error-correction capability of at least two. The conclusion a...
We construct infinite families of binary or ternary
linear complementary dual
(LCD) codes, which arise from some families of self-orthogonal codes associated with simplicial complexes or multi-variable functions. We find explicit criteria for self-orthogonal codes to preserve self-orthogonality after augmentation...
N. Mondal, Yoonjin Lee· Applicable Algebra in Engine...· 1 citation
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