Posted by: atri | October 12, 2007

## Lecture 20: Secret Sharing

Today we saw Shamir’s secret sharing scheme and its generalization to any linear code. In the next lecture, we will get back to the “traditional” coding theory stuff and start talking about explicit constructions and algorithms for codes (that hopefully will be close to achieving the capacity).

In class today Yang asked the following question: Given an ${[}n+1,k,d{]}_q$ code $C$ and its dual ${[}n+1,n+1-k,d^{\perp}{]}_q$ code $C^{\perp}$, why is $d+d^{\perp}< n+4$? (This was needed for the generalized secret sharing scheme to make sense). The argument that I have follows from two applications of the Singleton bound. By first applying it to $C^{\perp}$, we get that $d^{\perp}\le (n+1)-(n+1-k)+1=k+1$. Now by appying the Singleton bound to $C$, we get $k\le n+1-d+1=n-d+2$, which implies that $d^{\perp}\le n-d+3 as desired.