Projective MDS Codes Over GF(27)

MDS code is a linear code that achieves equality in the Singleton bound, and projective MDS (PG-MDS) is MDS code with independents property of any two columns of its generator matrix. In this paper, elementary methods for modifying a PG-MDS code of dimensions 2, 3, as extending and lengthening, in order to find new incomplete PG-MDS codes have been used over GF(27). Also, two complete PG-MDS codes over GF(27) of length 16 and 28 have been found.


Introduction:
Let = ( ) denote the Galois field with elements and let denote the characteristic of ( ). An [ , ]-code over is a -dimensional subspace of . The parameter is called the length of . The weight ( ) of a vector ∈ is the number of non-zero coordinates of . The minimum non-zero weight of all codewords in is called the minimum weight (Hamming distance) of and then an [ , ]code with minimum weight is called an [ , , ]code. From Singleton Bound Theorem, the parameter has maximum value − + 1, when and are given (1). The linear code that achieves equality in the Singleton bound is called, or MDS codes for short. The code which has minimum weight that correct errors that can be accrued is called -error correcting code and = ⌊ To any an [ , ]-code over , there is also another parameter = − called the redundancy, which represent the check digits added to the message to give protection against noise. The orthogonal complement of an [ , ]-code (the set of all vectors which are orthogonal to every vector in ), is called the dual code of with dimension − , and denoted by ⊥ . For more on linear codes, see (1).
Any linear code over with the three fundamental parameters, its length is , its dimension is , and its redundancy has a natural interpretation of each of these parameters. There are six basic modification techniques to linear codes depending on these three parameters. Each fixes one parameter and increases or decreases the other two parameters accordingly. These techniques are partitioned into three pairs and each member of a pair is the inverse process to the other as summarized below.  Since the redundancy of a code is its "dual dimension," each technique also has a natural dual technique.
The motivation of this research is based on two main ideas in the coding theory which can be summarized as follows: 1-It is well known algebraically; that two linear codes with same parameters will have the same efficiency if they are linearly isomorphism (equivalent).

2-
The fundamental problem to find codes with two properties: (i) reasonable information content (length is big enough), (ii) reasonable error handling ability.
A number of researchers worked on these two ideas recently in the sense of modification, for instance, see (2). Grassl (3) presented a new table with bounds to good codes not MDS code (small length and large minimum Hamming distance ) for 2 ≤ ≤ 9. Emami and Pedram (4) use punctured and shortened methods to construct codes (optimal Linear codes) with minimum value of for certain dimension and minimum Hamming distance . For further authors whose used shortening or puncturing structure of codes see; (5), (6) and the references therein. Some rehtr rtetarrhtr rr rt eraet r eht et traerrie t er'te rt a s' tar rret to achieve the singleton bound on the minimum distance; that is, code with maximum ability to correct errors (GM-MDS) (7), (8), (9), (10).
The main aim to this paper is to work with especial type of maximum distance separable (MDS) codes namely, projective MDS code (11) over 27 , since they provide the maximum protection against device failure for a given amount of redundancy; that is, the greatest error correcting capability (since error correcting capability is a function of minimum distance).To do that, an extending (dually, Lengthening) technique has been used.
The article is organized as follows. First section provided basic definitions and some properties of MDS and finite projective geometry. In second section, the inequivalent, incomplete projective MDS codes of dimension two have been constructed. Finally, in last section, the inequivalent, incomplete (complete) projective MDS codes of dimension three have been constructed, and three special complete MDS codes of lengths 16 and 28 have been founded using projective conic in the projective plane.

Definitions and Basic Properties
Any linear [ , ]-code can be defined by a ( × ) matrix or by a ( − ) × matrix whose entries from as defined below. Definition 1: (1) A generator matrix of an [ , ]code is a × matrix whose rows form a basis for . The standard form of a generator matrix is [ ]. A linear code for which any two columns of a generator matrix are linearly independent is called a projective code (PG-code). A linear code which cannot extend by adding columns to its generator matrix is called a complete code, otherwise it is called incomplete code.

Let
(2, ) denote the 2-dimensional projective space over (finite projective plane). Definition 3: (12) A non-singular plane quadric (form of degree two) in (2, ) is called a conic. A conic consists of + 1 points no three of which are collinear.
During the paper, the notation PG-MDS will briefly refer to a projective MDS code.

PG-MDS Code of Dimension 2 over
The technique used in this paper to check whether that two codes are projectively equivalent or not is as follows: The × matrices is called projectively equivalent to × matrices , and denoted by ≅ if there exist a non-singular × matrix such that matrix transformed to by performing the following operations: (i) make the last position of each row of , 0 or 1; (ii) a permutation of the columns on .  .
]. The matrix is of rank 2; that is, any two columns are linearly independents, so this gives the incomplete PG-MDS code with the parameters [3,2,2]. The 2 × 3 matrix can be extended in to 2 × 4 matrices, * by adding appropriate 25 columns [ 1 ] to from right side of , where ∈ 27 , = 1,2, … ,25, such that the rows of the new matrix * still linearly independent and any two columns is linearly independent. So, these 25 matrices, gives raise to 25 generated matrices * of PG-MDS codes. Among these 25 matrices, only 5 of them are non inequivalent as given the next theorem. To extend each matrix , an appropriate column [ 1 ] is added to for which does not belong to the first row of . So, there are ( − 3) possibilities for [ 1 ]; that is, 24 possibility. Therefore, by this way, 5( − 3) = 120 cods can be constructed. This procedure will be used to extend the codes in this paper.
In the next theorems, only the inequivalent codes are presented. and error correcting = 2 as given in Table 1.  New inequivalent, incomplete PG-MDS codes for fixed dimension = 2 and length 7 ≤ ≤ 14 over 27 can be constructed by means of a combinatorial computer search, and using the same technique in Theorem 2, 3 and 4. In the next theorem, the full details about these codes are given.

Conclusion:
In this paper the extending and lengthening are used to conclude the existence of incomplete, projective MDS codes of dimension two and three over the finite field of order twenty-seven. Where if = 2, codes of length , 4 ≤ ≤ 26 and distance , 3 ≤ ≤ 25 are founded. Also, if =3, codes of length , 4 ≤ ≤ 26 and distance , 2 ≤ ≤ 24 are founded. Two complete, projective MDS have been computed of dimension three and length sixteen.