Let $ { bf a} = [a_1, . . . , a_n]$ to be a sequence of $ n $ distinct elements taken in the finished body $ mathbb {F} _q $ and let $ { bf v} = [v_1, . . . , v_n]$ to be a sequence of n non-zero elements of $ mathbb {F} _q $. Let k $ to be a non-negative integer. Given that $ bf a $, $ bf v $ , and k $ in magma the order $ mbox {GRS} _k ({ bf a, v}) $ builds the generalized code of Reed – Solomon on $ mathbb {F} _q $. It's a $[n, k’]$ code as $ k $ leq $.

In addition, we consider the extension $ mbox {GRS} $ noted code $ mbox {GRS} _k ({ bf a, v}, infty) $. The code is given by

$$

mbox {GRS} _k ({ bf a, v}, infty) =

{(v_1f (a_1), …, v_nf (a_n), f_ {k – 1}): quad f (x) in mathbb {F} _q[x], quad deg (f) ≤ k – 1 },

$$

or $ f_ {k – 1} $ represents the coefficient of $ x ^ {k – 1} $ in $ f (x) $. It is known that the extension of the GRS code is still an MDS code. So,

$ mbox {GRS} _k ({ bf a, v}, infty) $ is a $ q- $ary $[n + 1, k, n − k + 2]$ $ mbox {MDS} $ code.

**My question:**

Suppose we built the code $ mbox {GRS} _k ({ bf a, v}) $ in magma. How to obtain $ mbox {GRS} _k ({ bf a, v}, infty) $ of $ mbox {GRS} _k ({ bf a, v}) $ in the Magma software?

*My essay:*

First I build the code $ C = mbox {GRS} _k ({ bf a, v}) $ in magma. Then by

by applying the command B = ExtendCode (C), I build an extension of C. But this command does not produce the code $ mbox {GRS} _k ({ bf a, v}, infty) $since the B code must be an MDS code, but in some examples, the ExtendCode (C) command generates a non-MDS code.

More precisely, suppose that two vectors $ { bf a} = [a_1, . . . , a_n]$ and $ { bf v} = [v_1, . . . , v_n]$ we built the $ mbox {GRS} _k ({ bf a, v}) $. Now, how to define two vectors $ { bf a} = [a’_1, . . . , a’_{n+1}]$ and $ { bf v} & # 39; = [v’_1, . . . , v’_{n+1}]$ of $ bf a $ and $ bf v $ such as

$ mbox {GRS} _k ({ bf a,, v $)) $ bean $[n+1,k]$ MDS code.

Thank you for all suggestions.