## Abstract

Let V be an rn-dimensional linear subspace of Z_{2}^{n}. 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^{-Ω(r2/log(1/r)+1n)}. We also answer a question of Ben-Or and show that if r > 1/2, then for every k, at most C_{r} · |V|/√n vectors of V have weight k. Our work draws on a simple connection between extremal properties of linear subspaces of Z_{2}^{n} and the distribution of values in short sums of Z_{2}^{n}-characters.

Original language | American English |
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Pages (from-to) | 497-522 |

Number of pages | 26 |

Journal | Combinatorica |

Volume | 22 |

Issue number | 4 |

DOIs | |

State | Published - 2002 |

### Bibliographical note

Funding Information:* Supported in part by grants from th e Israeli Academy of Sciences and th e Binational Science Foundation Israel-USA. † Th is work was done wh ile th e auth or was a student in th e Hebrew University of Jerusalem, Israel.