Related papers: Minimum Distance and the Minimum Weight Codewords …
A new construction for constant weight codes is presented. The codes are constructed from $k$-dimensional subspaces of the vector space $\F_q^n$. These subspaces form a constant dimension code in the Grassmannian space $\cG_q(n,k)$. Some of…
We present a tree-based construction of LDPC codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance, and perform well with iterative decoding. The construction involves enumerating a $d$-regular tree…
X-codes form a special class of linear maps which were originally introduced for data compression in VLSI testing and are also known to give special parity-check matrices for linear codes suitable for error-erasure channels. In the context…
In this article we investigate Berlekamp's negacyclic codes and discover that these codes, when considered over the integers modulo 4, do not suffer any of the restrictions on the minimum distance observed in Berlekamp's original papers.…
Goppa codes form an important class of alternant codes with wide applications in algebraic coding theory and code-based cryptography. Determining the true minimum distance of a Goppa code is a difficult problem. In this paper, we provide a…
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 Fp or Fp2 ; or over even degree…
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then…
The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem…
We comprehensively study weighted projective Reed-Muller (WPRM) codes on weighted projective planes $\mathbb{P}(1,a,b)$. We provide the universal Gr\"obner basis for the vanishing ideal of the set $Y$ of $\mathbb{F}_q$--rational points of…
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy…
In 1970 Delsarte, Goethals and Mac Williams published a seminal paper on generalized Reed-Muller codes where, among many important results, they proved that the minimal weight codewords of these codes are obtained through the evaluation of…
We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close…
We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a…
In this paper, we study the minimum distances of binary linear codes with parity check matrices formed from subset inclusion matrices $W_{t,n,k}$, representing $t$-element subsets versus $k$-element subsets of an $n$-element set. We provide…
Affine Cartesian codes were first discussed by Geil and Thomsen in 2013 in a broader framework and were formally introduced by L\'opez, Renter\'ia-M\'arquez and Villarreal in 2014. These are linear error-correcting codes obtained by…
All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact…
We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form $[\mathbf{C} \vert \mathbf{M}]$ where $\mathbf{C}$ is circulant and of column weight $2$, and…
A linear code over $\mathbb{F}_q$ with the Hamming metric is called $\Delta$-divisible if the weights of all codewords are divisible by $\Delta$. They have been introduced by Harold Ward a few decades ago. Applications include subspace…
We design a heuristic method, a genetic algorithm, for the computation of an upper bound of the minimum distance of a linear code over a finite field. By the use of the row reduced echelon form, we obtain a permutation encoding of the…
Few-weight codes have been constructed and studied for many years, since their fascinating relations to finite geometries, strongly regular graphs and Boolean functions. Simplex codes are one-weight Griesmer $[\frac{q^k-1}{q-1},k…