📝 SummarXiv

Abstract

Bounds on linear codes play a central role in coding theory, as they capture the fundamental trade-off between error-correction capability (minimum distance) and information rate (dimension relative to length). Classical results characterize this trade-off solely in terms of the parameters $n$, $k$, $d$ and $q$. In this work we derive new bounds under the additional assumption that the code contains a nonzero codeword of weight $w$.By combining residual-code techniques with classical results such as the Singleton and Griesmer bounds,we obtain explicit inequalities linking $n$, $k$, $d$, $q$ and $w$. These bounds impose sharper restrictions on admissible codeword weights, particularly those close to the minimum distance or to the code length. Applications include refined constraints on the weights of MDS codes, numerical restrictions on general linear codes, and excluded weight ranges in the weight distribution. Numerical comparisons across standard parameter sets demonstrate that these $w$-aware bounds strictly enlarge known excluded weight ranges and sharpen structural limitations on linear codes.