---
tags: Assignments
---

# Assignment 3

:::info
Due Friday September 16
:::

1. Let $A$ be the incidence graph of a $2-(v, k, \lambda)$ design with $b$ blocks and $r$ blocks incident with each point. Express the spectrum of $A$ in the design parameters $v$, $k$, $\lambda$, $b$ and $r$. Give reasons for your answer. __(5 marks)__
2. Let $C$ be the code over $\mathbb{Z}_p$ generated by the characteristic vectors of the blocks of a design $D$ with $v$ points. Suppose that there is an $s$ such that any two (not necessarily distinct) blocks meet in a number of points that is congruent to $s\pmod{p}$. Show that $\dim C \leqslant (v+1)/2$. __(6 marks)__
3. Consider the linear code $C=\{(0,0,0),(1,0,0),(0,1,0),(0,0,1),(1,1,1),(1,1,0),(0,1,1),(1,0,1)\}.$ Define relations $R_i$ on $C$ by $u\,R_i\,v\iff d(u,v)=i$ (where $i=0,1,2,3$). Show that the $R_i$ define an association scheme on $C$ by computing the intersection matrices of it. __(4 marks)__