## Abstract

We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two n × n matrices over finite fields. In particular we obtain the following results: 1. We show that the number of product gates in any bilinear (or quadratic) circuit that computes the product of two n × n matrices over GF(2) is at least 3n^{2} - o(n^{2}). 2. We show that the number of product gates in any bilinear circuit that computes the product of two n × n matrices over GF(q) is at least (2.5 + 1.5/q^{3}-1)n^{2}-o(n^{2}). These results improve the former results of [N. H. Bshouty, SIAM J. Comput., 18 (1989), pp. 759-765; M. Bläser, Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 1999, pp. 45-50], who proved lower bounds of 2.5n^{2} - o(n^{2}).

Original language | English |
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Pages (from-to) | 1185-1200 |

Number of pages | 16 |

Journal | SIAM Journal on Computing |

Volume | 32 |

Issue number | 5 |

DOIs | |

State | Published - Aug 2003 |

Externally published | Yes |

## Keywords

- Linear codes
- Lower bounds
- Matrix product