Linear Codes Arise From New Complete (n,r)-arcs in PG(2,29)

This paper presents the recently-discovered linear [n,3,d] codes over PG(2,29) that arises from a complete (n,r)-arcs which the paper[12] presented it for the first time. The aim of this paper is to formulate the recently discovered upper bounds and lower bound for (n,r)-arcs as bounds that will look familiar to coding theorists.New two lists in this paper appeared, the first list of 15 codes arranged from[164,3,156]-code up to [704,3,678]-code, the second list of 27 codes arranged from [28,3,25]-code up to [776,3,747]-code, they are appeared for the first time in this paper, all of these codes we can call them as complete codes as thier definition in this paper, they belong to the class of error-correcting codes (ECC). In this paper I made a computer programs to construct these new codes with Random Greedy Construction method (RGC) which is mentioned in [13]


Introduction ( Linear codes and Error-Correcting Codes )
Communicating information from one person to another is, of course, an activity that is as old as mankind. The (mathematical) theory of the underlying principles is not so old. It started in 1948, when C.E. Shannon gave a formal description of a communication system and, at the same time, also introduced a beautiful theory about the concept of information, including a good measure for the amount of information in a message. The theory of error detecting and correcting codes (ECC) is that branch of engineering and mathematics which deals with the reliable transmission and storage of data. Information media are not 100% reliable in practice, in the sense that noise (any form of interference) frequently causes data to be distorted. To deal with this undesirable but inevitable situation, some form of redundancy is incorporated in the original data. With this redundancy, even if errors are introduced (up to some tolerance level), the original information can be recovered, or at least the presence of errors can be detected. We saw in class how adding to the original message the parity bit or the arithmetic sum allows the detection of a (certain type of) error. However, that kind of redundancy doesn't allow for the correction of the error. Error-correcting codes do exactly this: they add redundancy to the original message in such a way that it is possible for the receiver to detect the error and correct it, recovering the original message. This is crucial for certain applications where the re-sending of the message is not possible (for example, for interplanetary communications and storage of data). The crucial problem to be resolved then is how to add this redundancy in order to detect and correct as many errors as possible in the most efficient way. Error-correcting codes are particularly suited when the transmission channel is noisy. This is the case of wireless communication. Nowadays, all digital wireless communications use error-correcting codes. This paper sets new codes that are not known until now, it's codes appeared from (n,r)-arcs in the finite projective plane PG (2,29), this information in this research finds new correcting codes that were not known before, so the benefit of this paper is to use it's codes in transmitting security information among large distance without using the normal used codes that may be exposed it's security .

Preliminary
At first I must give some definitions, a linear [n,k,d] code over finite field Fq is a k-dimensional subspace of the n-dimensional vector space V(n,q) over Fq such that d is the smallest number of positions in which two different elements of the code differ [6]. Let PG(2,q) be a finite projective plane Π of order q , where q= 1 ,  h p h this plane consists of q 2 +q+1 lines and the same number of points , q+1 points on every line and q+1 lines passing through every point [8]. An (n,r)-arc is a set K of n points in PG(2,q) with at most r points on a line but there are no r+1 or more on any line [10], an (n,r)-arc is called complete, if it is not contained in an (n+1,r)arc [11]. A line L of the plane containing precisely i points of K, called an i-secant .Let Ti denote the total number of i-secants to K in PG(n,q). Hamming distance d on F n q× F n q is given by d(x;y) = #{i: xi ≠ yi}, where x = (x1,…, xn) and y = (y1,… yn). The weight of x is defined by w(x) := d(x,o) ,where o := (0,…,0) [9]. The minimum distance of a code C  F n q is given For a linear code C  F n q we have d(C) = min{w(x): xC|{0}}.Let C  F n q be a linear code of dimension k,a generator matrix of C is a k × n matrix whose rows form an Fq-base of C. Let C  F n q be a code, the dual code of C is the code C ⊥ defined by C ⊥ :={x F n q: is the usual bilinear form on F n q× F n q . Note that C ⊥ is indeed a linear code. For x  F n q , let x t denote its transpose.

Lemma[3]
Let C  F n q a linear code of dimension k and M a generator matrix of C, Then (1) C ⊥ = { x  F n q : Mx t =0}; (2) C ⊥ has dimension n-k.

Corollary[3]
: Let C be a linear code and H a generator matrix of C ⊥ . Then: (1) C = (C ⊥ ) ⊥ ; (2) C={ x  F n q :Hx t =0}. The redundancy of a k-dimensional linear code in F n q is n -k. A parity check matrix of a linear code is any generator matrix of its dual. (2) We have d(C) = min{wZ + |  w columns Fq-linearly dependent in H}.

Corollary[5] (Singleton Bound)
For an Fq-linear code of length n, dimension k and minimum distance d,

Definition[3]
: Let C be an [n,k,d]-code, when the Singleton defect s(C) = 1, C is said to be an Almost MDS code (AMDS code for short).

Proposition[3]:
The dual code of an MDS code is also MDS.

Relation between linear codes and (n,r)-arcs
Write out the points of the (n,r)-arc K as columns of a matrix G, then form the code C as linear combinations of the rows of G. So, C is an [n,3,d]code. What is d? Think of it in this way. The rows of G are as follows: r1 = x1 x2 ... xn r2 = y1 y2 ... yn r3 = z1 z2 ... zn If the line L with equation ax + by + cz =0 contains exactly s points of K, then the codeword ar1 + br2 + cr3 has weight n -s. This is because, if ax + by + cz is zero for the points P1,P2,...,Ps, it is not zero for the other n-s points of K.So, this implies that, since any line contains at most r points, the weight of a codeword is at least n-r. Since some line contains exactly r points, so the minimum weight d = n-r. Further, if you count the numbers Ti for K, where Ti is the number of lines meeting K in exactly i points, then the numbers (q -1)Ti give the weight distribution of the code. [4] Definition: If the (n,r)-arc is complete then we call the corresponding code for it a complete code.
Hence if one can get the matrix G so, he gets the code C (where G is it's generator matrix).For example if our arc contains from the following points {0,2,3,869,870} from the points of PG(2,29) so, the generator matrix G will be written as the coordinates of each point contains from the same arc {0 2 3 869 970} as written here on the finite field F29 . For simplicity denote to the points by its order in the finite field without typing the coordinates for each one.

What is RGC-Method ?
When one wants to construct an object with certain structural constrains such as packings, covers, graphs without certain small subgraphs and arcs in a plane, random greedy construction is considered as a natural way to generate it : Randomly order all possible elements of the desired object and select each of them one by one in the order if and only if it together with already selected ones cause no conflict, i.e. no violation to the given constrains. Here we mean by "select" that we choose and permanently add it to the desired object being constructed. We may discard at each step all elements that cause any conflict with already selected ones and then randomly select a non-discarded one. This is an equivalent construction and will be called the Random Greedy Construction (RGC). For example, the RGC of a complete arc is the following. Initially, the arc being constructed is empty. At each step, discard all points contained in any secant of already selected points and select one non-discarded point uniformly at random. Then the set of all selected points is a complete arc. In many cases, it is believed that the RGC yields an almost optimal desired object. [13]

Second List of ECC
It depends on the latest appeared minimum bounds for (n,r)-arcs which were not appeared even in [2].So, I can set them as follows: The following codes are now exist :   n  k  d  type  28  3  25  Complete code  46  3  42  Complete code  62  3  57  Complete code  82  3  76  Complete code  100  3  93  Complete code  125  3  117  Complete code  152  3  143  Complete code  177  3  167  Complete code  203  3  192  Complete code  230  3  218  Complete code  254  3  241  Complete code  282  3  268  Complete code  310  3  295  Complete code  337  3  321  Complete code  363  3  346  Complete code  390  3  372  Complete code  422  3  403  Complete code  453  3

Two samples
For example the first two codes from the second list have the following generator matrices G1 and G2 respectively :

Conclusion :
This research finds new correcting codes that are not known before, so the benefit of this paper is to use it's codes in transmitting security informations among large distance without using the normal used codes that may be exposed it's security .