arXiv subjects

Romar dela Cruz

Publications and source records attributed to Romar dela Cruz.

5 recordsLinked to original sources

arXiv2020

On the maximum number of minimal codewords

Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number are presented. We determine the exact values for the case of linear codes of dimension $k$ and length $k+2$ and for small values of the length and dimension. We also give a formula for the number of minimal codewords of linear codes of dimension $k$ and length $k+3$.

cs.IT↗

arXiv2019

On the minimum number of minimal codewords

We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.

math.CO↗

arXiv2019

Majority-logic Decoding with Subspace Designs

Rudolph (1967) introduced one-step majority logic decoding for linear codes derived from combinatorial designs. The decoder is easily realizable in hardware and requires that the dual code has to contain the blocks of so called geometric designs as codewords. Peterson and Weldon (1972) extended Rudolphs algorithm to a two-step majority logic decoder correcting the same number of errors than Reed's celebrated multi-step majority logic decoder. Here, we study the codes from subspace designs. It turns out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their majority logic decoding complexity is sometimes drastically improved.

math.CO↗

arXiv2012

The maximum number of minimal codewords in long codes

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide lower bounds. In this paper we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with $p$ vertices and $q$ edges can have only slightly more that $2^{q-p}$ cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than $2p+3\log_2(3p)$ edges. We also conclude that an Eulerian (even) graph has at most $2^{q-p}$ cycles unless the graph is a subdivision of a 4-regular graph that is the edge-disjoint union of two Hamiltonian cycles, in which case it may have as many as $2^{q-p}+p$ cycles.

cs.IT↗

arXiv2010

An extension of Massey scheme for secret sharing

We consider an extension of Massey's construction of secret sharing schemes using linear codes. We describe the access structure of the scheme and show its connection to the dual code. We use the $g$-fold weight enumerator and invariant theory to study the access structure.

cs.IT↗