Minimal codewords over finite fields derived from certain graphs

초록

Throughout this paper, we explore the number of non-equivalent minimal codewords of linear codes derived from certain graphs. We propose a lower bound on the number of non-equivalent minimal codewords over Fq${\mathbb{F}}_q$ associated with graphs of diameter 2. Beyond diameter 2, we also determine the number of non-equivalent minimal codewords over Fq${\mathbb{F}}_q$ for graphs with arbitrary diameter. To achieve this, we study n-cycles and the row spaces generated by some rows from the generator matrix of linear codes. Primarily, our focus is on the number of non-equivalent minimal codewords, and we also provide precise construction methods for identifying minimal codewords in linear codes. To support our results, we present some examples in this work.

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