Certain Types of Linear Codes over the Finite Field of Order Twenty-Five

The aim of the paper is to compute projective maximum distance separable codes, -MDS of two and three dimensions with certain lengths and Hamming weight distribution from the arcs in the projective line and plane over the finite field of order twenty-five. Also, the linear codes generated by an incidence matrix of points and lines of ( ) were studied over different finite fields


Introduction
Let ( ) denotes the Galois field of elements, is a prime power, is a plus point at infinity, and is the vector space of row vectors of length with entries in . Let ( ) be the corresponding projective space of dimension . As a special case, ( ) and ( ) are called projective line and projective plane, respectively. The points ( ) of ( ) are the one dimensional subspaces of . In ( ), the number of points is ( ) ( ) ( ) ⁄ and the number of lines is . An ( )-arc with is a set of points of a projective space, such that most points are on the hyperplane, but with at least one set of points are on the hyperplane. In the line, ( )-arc is just an -set; that is, a set of distinct points. An ( )-arc is called complete if it is maximal with respect to inclusion; that is, there is no an ( )-arc containing . The maximum size of an ( )-arc in ( ) is denoted by ( ). In 1947, Bose [1] proved that ( ) ( ) In the finite projective line, the value of ( ) is just ISSN: 0067-2904

Al-Zangana and Shehab
Iraqi Journal of Science, 2021, Vol. 62, No. 11, pp: 4019-4031 4444 Definition 1. A conic in ( ) is the set of rational points of a homogenous nonsingular form of degree two over . Bose showed that: an ( ( ) )-arc in ( ) odd, is just the conic, and that the conic plus its nucleus (the intersection point of its tangents) is an ( ( ) )-arc in ( ) even. The points ( ) of the projective line ( ) are identified by by sending the points ( ) to ⁄ if and to if . The relation between the conic ( ) and exists by sending each point of to ( ) point on the conic . For details and basic results on the projective space and the essential subsets of the projective space, see [2]. The Hamming weight of a vector is the number of non-zero coordinates of denoted by ( ). A -ary , --code over is a -dimensional subspace of , all of whose non-zero vectors (codewords) have a weight of at least ( ). A -ary , --code that corrects ⌊ ⌋ errors is called -error correcting code, where ⌊ ⌋ denotes the floor function. Let denotes the number of codewords with Hamming weight in a code of length . The sequence ( ) is called the weight distribution of the code . The dual code of -ary , --code over , denoted by , is defined by Any -ary , --code can be defined by a ( ) matrix , -(standard form), where is a nonsingular ( ) matrix with entries from , called the generator matrix, whose rows form a basis. Also, the dual code can be defined by a ( ) matrix [ ( ) ]. Two linear codes are isomorphic (equivalent) if the generator matrices are equivalent after doing a sequence of row (column) operations. A sphere-packing bound of a -ary , --code over is

{∑ . / ( ) }
A code which achieves the sphere-packing bound is called a perfect code, see [3]. Definition 2 [4]. A -ary , --code over at (the maximum value of ) is called a maximum distance separable code, or MDS code for short. The code is called projective if the columns of a generator matrix are pairwise linearly independent and denoted by -MDS. Theorem 3 [4] A -ary , --code over is MDS if and only if its dual is MDS; that is, ( ) if and only if ( ) . Therefore, A -ary , --code over is -MDS if and only if its dual is -MDS, since the standard generator matrix of both are depending on the base matrix . It is well known that there is equivalence between the existence of a -MDS and an arc in the projective space, where this equivalence comes from the fact that the matrix in which each column is a point of an arc has formed a generator matrix of -MDS. The full prove of this relation is presented elsewhere [4] and the statement of the theorem is as follows.  (1) and (2), see [3].
Ezerman et al. [5] determined the weight spectra of certain linear MDS codes, namely those that satisfy the MDS Conjecture. Alderson [6] discussed the weight distribution of MDSary , --code and showed that all weights from to are realized. One of the important questions for a code with parameters and , is: how many nonisomorphic codes are there having these parameters? Many researches discussed this question directly by working on the code, see for example [7,8], or indirectly through projective space, both in general cases and for a certain , see for example [9,10,11]. The first objective of this paper is to present a class of non-isomorphic error-correcting -MDS codes over of two and three dimensions with their weight distributions. The second objective is to construct linear codes from the incidence matrix of lines and points of ( ) by giving details of generator matrices over distinct finite fields. The GAP programming was used to perform the calculations required for achieving the desired results [12].

Non-Isomorphic Error-Correcting
-MDS Codes over Al-Zangana and Shehab [13] gave full details of the classification of projectively inequivalent -subsets in the projective line over such that each -subset contains the standard frame ( ) * +. These results are summarized in Table 1. Let denotes the number of projectively inequivalent -subsets of ( ). Theorem 5. Over , the non-isomorphic -MDS codes with parameters , and no zero weight distributions are listed in Table 2. Here ̂ denotes the number of non-isomorphic -MDS codes of specific parameters. Proof. First of all, since each -subset computed in [13] contains the points of the standard frame, then the constructed ( ) matrix from the points of -subset will be in a standard form and the second row of takes the form ; that is, ,and a ( ) matrix has now zero coordinate in each row (column) vector. According to the construction of points of the projective line, the second coordinate is 1 and, hence, the second row of is always a vector with one in each coordinate. Hence, it is enough to give the first row of the matrix to refer to the generator matrix. Secondly, from Theorem 4, everysubset formed a -MDS -, --code. For each , the GAP program was used to compute the weight distributions , . Let be the primitive element of . . , the first rows of a one generating matrix are written below, since there is no enough space to write all here.

st row of generating matrix
The complement subset of each -subset formed an ( )-subset of ( ). Therefore, the number of inequivalent ( )-subsets and -subsets of ( ) is equal. Thus, the number of non-isomorphic -MDS codes with length equal to and dimension is equal to the number of non-isomorphic -MDS codes with length and dimension , where . The number of non-isomorphic -MDS codes with lengths and dimension is one, since all the -sets are equivalents. Also, there is only one non-isomorphism -MDS code of length and dimension , since the -subset of ( ) is just the line. Corollary 5. Over , the dual codes of the -MDS codes with parameters ̂ shown in Table 2, formed -MDS codes with dimension and . Proof. From Theorem 3, each dual code of the -MDS -ary , --code over formed -MDS -ary , --code and with . Since the dual code of is , then the number of non-isomorphic code for certain length is ̂, as in Table 2. The weight distributions ( ) of for fixed are as listed in Table 3. Al-Zangana and Shehab [14] proved that there are eight inequivalent -arcs and 365 inequivalent -arcs in the projective plane over through the standard frame ( ) * +. The corresponding PG-MDS codes to these arcs are summarized in the following theorem. transformed to after dividing the first, second, and third columns of by and applying some permutations in rows and columns. Thus, is equivalent to .

Codes from Incidence Matrix
The incidence matrix ( ) of points and -dimensional projective subspaces in the projective space ( ) , prime, , is defined as the matrix whose rows are indexed by the -spaces of ( ), , and whose columns are indexed by the points of ( ), and with the entry { Clearly, the dimension of is ( ) ( ). For more details, see [15,16].
It is known that the rows of the matrix generate a -ary , --code over a field . This code is normally denoted by ( ), and by ( ) if and .
The minimum weight of ( ) is which is provedin by giving the general case for that. Therefore, ⌊ ⌋ ⌊ ⌋ ⌊ ⌋.
Over , The incidence matrix ( ) of points and lines in the projective space ( ) was computed. An algorithm was executed with GAP program to compute the generator matrices of linear codes from over several finite fields. The results are summarized below.

The matrix
is given by identifying each row, , by a non-zero position, as shown below.
The dual code of -ary , --code is cyclic, -ary , --code. The coefficient of the generator polynomials which are of degree is just 's. When , the weight of each row of the generator matrix is and its covering radius is . Since , then this code is perfect.

Conclusions
Over the finite field of order twenty-five, using ideas of arcs in the projective space, many non-isomorphic projective MDS were found. Also, with incidence matrix idea of points and lines in the projective space, many other linear perfect (non-perfect) codes were founded. The most important property of the rows of incidence matrix is that each -th row is just circulate to the ( )-th row, except the last row. The best linear code that can be constructed from the incidence matrix is when it is taken over , since it will have a Hamming weight , while over the others that are of order distinct from it behaves like a trivial code. Also, when the matrix is taken over , a perfect code is founded.