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Small weight codewords of projective geometric codes. (English) Zbl 1512.94118

Summary: We investigate small weight codewords of the \(p\)-ary linear code \(\mathcal{C}_{j , k}(n, q)\) generated by the incidence matrix of \(k\)-spaces and \(j\)-spaces of \(\mathrm{PG}(n, q)\) and its dual, with \(q\) a prime power and \(0 \leq j < k < n\). Firstly, we prove that all codewords of \(\mathcal{C}_{j , k}(n, q)\) up to weight \(\left( 3 - \mathcal{O} ( \frac{ 1}{ q} )\right) \begin{bmatrix} &k + 1\\ &j + 1 \end{bmatrix} _q\) are linear combinations of at most two \(k\)-spaces (i.e. two rows of the incidence matrix). As for the dual code \(\mathcal{C}_{j , k} ( n , q )^\bot \), we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight codewords (2) to the case \(\mathcal{C}_{0 , 1} ( n , q )^\bot \). This implies the solution to both problem (1) and (2) if \(q\) is prime and the solution to problem (1) if \(q\) is even.

MSC:

94B05 Linear codes (general theory)
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
94B65 Bounds on codes

Citations:

Zbl 1478.94141
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References:

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