Linear Codes and Character Sums

Let V be an rn-dimensional linear subspace of \( Z^{n}_{2} \). Suppose the smallest Hamming weight of non-zero vectors in V is d. (In coding-theoretic terminology, V is a linear code of length n, rate r and distance d.) We settle two extremal problems on such spaces.

First, we prove a (weak form) of a conjecture by Kalai and Linial and show that the fraction of vectors in V with weight d is exponentially small. Specifically, in the interesting case of a small r, this fraction does not exceed \( 2^{{ - \Omega {\left( {\frac{{r^{2} }} {{\log {\left( {1/r} \right)} + 1}}n} \right)}}} \).

We also answer a question of Ben-Or and show that if \( r > \frac{1} {2} \), then for every k, at most \( C_{r} \cdot \frac{{{\left| V \right|}}} {{{\sqrt n }}} \) vectors of V have weight k.

Our work draws on a simple connection between extremal properties of linear subspaces of \( Z^{n}_{2} \) and the distribution of values in short sums of \( Z^{n}_{2} \)-characters.

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Correspondence to Nathan Linial*.

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* Supported in part by grants from the Israeli Academy of Sciences and the Binational Science Foundation Israel-USA.

† This work was done while the author was a student in the Hebrew University of Jerusalem, Israel.

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Linial*, N., Samorodnitsky†, A. Linear Codes and Character Sums. Combinatorica 22, 497–522 (2002). https://doi.org/10.1007/s00493-002-0003-y

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AMS Subject Classification (2000):

  • 94B65
  • 05D05