Let be a linear code with parameters Namely, is a -subspace of with dimension and the minimum (Hamming) distance We have the Singleton bound is called MDS code if The dual code of is defined by
where for and is the usual inner product in . is a linear code with parameters where If is called self-dual. For self-dual code , the length of the code should be even by
MDS self-dual codes have attracted a lot of attention in recent years by their theoretical interests in coding theory, many applications in cryptography and combinatorics. For such codes, and are determined by the length . One of the basic problems is that for a fixed prime power which even number can be the length of an MDS self-dual code over finite field ?
This problem has been solved in the case by Grassl and Gulliver . From now on, we consider where
is an odd prime number and. Several families of MDS self-dual codes over have been constructed with length satisfying certain conditions by using generalized Reed-Solomon (GRS for short) codes and extended generalized Reed-Solomon (EGRS for short) codes -,,-,, orthogonal designs [8, 14], extended cyclic duadic codes and negacyclic codes . Roughly speaking, the first approach is to look for the GRS codes and EGRS codes as candidates of MDS codes, then to find sufficient conditions satisfied by length such that the codes are self-dual. The last two approaches are to look for the self-dual codes given by orthogonal designs and (nega-)cyclic codes and select ones being MDS codes. A table of MDS self-dual codes over is provided in  for length and odd prime number
In this paper, we focus on the first approach to consider the MDS codes given by GRS and EGRS codes. L.Jin, C.Xing  and H.Yan  present basic results on necessary and sufficient conditions for the GRS codes and EGRS codes being self-dual respectively. By using these conditions, several families of MDS self-dual codes with various length have been constructed [2, 3, 9, 12, 15] with careful computations.
In Section 2, we introduce the basic results given in  and , but we provide an unified point of view on the constructions given there. In Section 4, we select some known results stated and proved with our approach. In Section 5, we construct new MDS self-dual codes with our approach. In Section 6, we make conclusion and raise two open problems.
2. Constructions of MDS Self-dual codes via GRS Codes
2.1. (Extended) Generalized RS Codes
Let be a subset of with distinct elements (so that ), be nonzero elements in (not necessarily distinct), For the GRS code is defined by
This is an MDS (linear) code over with parameters The extended GRS code is defined by
where is the coefficient of in This is also an MDS code over with parameters
For we denote
Let be the quadratic (multiplicative) character of Namely, for ,
Let be distinct elements in
(1) () Suppose that is even. There exists such that the (MDS) code is self-dual if and only if all are the same.
(2) () Suppose that is odd. There exists such that the (MDS) code is self-dual code with length if and only if for all
Let be the set of all even number such that there exists MDS self-dual code over with length . Let and be the set of all even number such that there exists MDS self-dual code over with length constructed by generalized RS code (Theorem 2.2 (1)) and extended generalized RS code (Theorem 2.2 (2)) respectively. Namely,
We have As a direct consequence of Theorem 2.2, we have
() Suppose that and where is an odd prime number and Then for any even number if and if Therefore for all even number
3. Binomials and Linearized Polynomials
In general case, we should choose a subset of and compute carefully to verify the conditions stated in Theorem 2.2. The following simple fact can be used to simplify the computation of in many cases. For a polynomial we denote the derivative of by
(1) Let be a subset of Then for any
(2) Let and be disjoint subsets of Then for
(3) Let be a monic polynomial of degree in Assume that for each has exactly distinct solutions in Namely, the size of is for all Let Then for we have where
(1) From we get, for
(2) From we get
For and then
(3) For is the set of zeros of Namely, We get
where Then from we get, for
Let . In many known results the set is chosen as a subgroup of where or a coset of in or an union of coset For is a binomial and Another candidate of is -subspace of where is a subfield of a coset or an union of cosets. It is well-known that
is a -linearized polynomial in and where
Let be a subfield of A -linearized polynomial in has the following form
Let be a -linearized polynomial in . The mapping
is -linear. Namely, for
Thus the kernel and image of
are -subspaces of
For more facts on linearized polynomials we refer to the book , Section 3.4.
4. Review on Some Known Results
In this section, we select few known results on MDS self-dual codes constructed via GRS codes and EGRS codes. We present simple proofs with the approach illustrated in Section II. For more complete list of known MDS self-dual codes we refer to the table in  and .
() Let and Then for each if is even and if is odd.
Let for Then is a subfield of and From we know that We take a -subspace of with Then and
is a -linearized polynomial in Take then For any Therefore are distinct cosets of in We take a subset of with size Let and Then and
where Thus for each and we have and by Lemma 3.1 (3),
where and then
() Let Then for all even and we have
Take in Theorem 4.1. ∎
In Theorem 4.1, is chosen as an union of cosets of a subspace of Now we consider being an union of cosets of a (cyclic) subgroup of Let then is a subgroup of with All cosets of in are the -th cyclotomic classes
Let be distinct cosets of in Then
In , is taken as a subset of for Firstly we should determine that how many elements in can be taken such that the cosets are distinct.
Let where and Let be a subset of where are distinct module Then are distinct cosets of in if and only if are distinct module Particularly, if there exist elements in such that are distinct.
Therefore if and only if ∎
Now suppose that be a subset of such that are distinct module Let Then
is an union of cosets of in and For each by (4.1), we get
From we get
where Therefore where
With above preparation we introduce the following result which appear in  but with slight different statement.
Let be a power of a prime number and
(I) Assume that is even and