Paper 2, Section II, H

Coding and Cryptography
Part II, 2009

Describe the construction of the Reed-Miller code RM(m,d)R M(m, d). Establish its information rate and minimum weight.

Show that every codeword in RM(d,d1)R M(d, d-1) has even weight. By considering xy\mathbf{x} \wedge \mathbf{y} with xRM(m,r)\mathbf{x} \in R M(m, r) and yRM(m,mr1)\mathbf{y} \in R M(m, m-r-1), or otherwise, show that RM(m,mr1)RM(m,r)R M(m, m-r-1) \subseteq R M(m, r)^{\perp}. Show that, in fact, RM(m,mr1)=RM(m,r).R M(m, m-r-1)=R M(m, r)^{\perp} .