Posted by: atri | October 8, 2007

## Lecture 18: Johnson Bound

Today we stated and proved the Johnson bound (for $q=2$). Next lecture, we will use the Johnson bound to the prove the Elias-Bassalygo upper bound on the rate of a code (for a fixed distance).

I mentioned in class that the Johnson bound is tight in the sense that there exist linear codes that have super-polynomially many codewords in some Hamming ball of radius slightly larger than what is dictated by the Johnson bound. This result was established in the paper by Guruswami titled Limits to List Decodability of Linear Code (STOC02) modulo a number theoretic conjecture. The unconditional result was proved in a follow-up paper by Guruswami and Shparlinski titled Unconditional Proof of Tightness of Johnson Bound (SODA03).