Classical Goppa codes

Abstract

Let us recall that in (4.4) aBCHcode was defined as the set of words (c₀, c₁, ... , c_n-1) ∈ IFⁿ _q such that c₀ + c₁(β^j) + c₂(β^j)² + ... + c_n-1(β^j)^n-1 = 0 where ß is a primitive n^th root of unity and 1 ≤ j < d. Here d is the designed distance. We can rewrite this as follows:

$$\begin{array}{*{20}{c}} \hfill {({{z}^{n}} - 1)\sum\limits_{{i = 0}}^{{n - 1}} {\frac{{{{c}_{i}}}}{{z - {{\beta }^{{ - i}}}}} = \sum\limits_{{i = 0}}^{{n - 1}} {{{c}_{i}}\sum\limits_{{k = 0}}^{{n - 1}} {{{z}^{k}}{{{({{\beta }^{{ - i}}})}}^{{n - 1 - k}}} = } } } } \\ \hfill { = \sum\limits_{{k = 0}}^{{n - 1}} {{{z}^{k}}} \sum\limits_{{i = 0}}^{{n - 1}} {{{c}_{i}}{{{({{\beta }^{{k + 1}}})}}^{i}} = {{z}^{{d - 1}}}p(z),} } \\ \end{array}$$

i.e.