ON THE MINIMUM LENGTH OF SOME LINEAR CODES OF DIMENSION 6

Abstract

For $q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2$, we prove the non-existence of a $[g_q(6,d),6,d]_q$ code and we give a $[g_q(6,d)+1,6,d]_q$ code by constructing appropriate 0-cycle in the projective space, where $g_q (k,d)={{\sum}^{k-1}_{i=0}}{\lceil}\frac{d}{q^i}{\rceil}$. Consequently, we have the minimum length $n_q(6,d)=g_q(6,d)+1\;for\;q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2\;and\;q{\geq}3$.

Keywords

• Griesmer bound;
• linear code;
• 0-cycle;
• minimum length;
• projective space

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References

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Cited by

1. DETERMINATION OF MINIMUM LENGTH OF SOME LINEAR CODES vol.26, pp.1, 2013, https://doi.org/10.14403/jcms.2013.26.1.147