The subfield codes of hyperoval and conic codes

Abstract

Hyperovals in PG(2, GF(q)) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in PG(2, GF(q)) are equivalent to [q + 2, 3, q] MDS codes over GF(q), called hyperoval codes, in the sense that one can be constructed from the other. Ovals in PG(2, GF(q)) for odd q are equivalent to [q + 1, 3, q − 1] MDS codes over GF(q), which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the p-ary subfield codes of the conic codes. The weight distributions of these subfield codes and the parameters of their duals are determined. As a byproduct, we generalize one family of the binary subfield codes to the p-ary case and obtain its weight distribution. The codes presented in this paper are optimal or almost optimal in many cases. In addition, the parameters of these binary codes and p-ary codes seem new.Conversely, the column vectors of a generator matrix of any MDS [q + 2, 3, q] code over GF(q) form a hyperoval in PG(2, GF(q)). Thus constructing hyperovals in PG(2, GF(q)) is equivalent to constructing [q + 2, 3, q] codes over GF(q). Therefore, every [q + 2, 3, q] code over GF(q) is called a hyperoval code.A conic in PG(2, GF(q)) is a set of q + 1 points of PG(2, GF(q)) that are zeros of a nondegenerate homogeneous quadratic form in three variables. It is known that a conic is an oval in PG(2, GF(q)) and an oval in PG(2, GF(q)) is a conic if q is odd [2]. Hence, conics and ovals in PG(2, GF(q)) are the same when q is odd. Let q be odd. Define O = {(x 2 , x, 1) : x ∈ GF(q)} ∪ {(1, 0, 0)}.

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

“…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.…”

“…Therefore, the Hamming weight of this code is 1. Over the field , the Hamming weights of vectors in the generating matrix take the values 1, 2, 3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,…”

Optimal Binary Linear Codes From Maximal Arcs

“…By definition, the dimension k ′ of C (q) satisfies k ′ ≤ mk. To the best of our knowledge, the only references on subfield codes are [4], [5], [11], [14]. Recently, some basic results about subfield codes were derived and the subfield codes of ovoid codes were studied in [11].…”

“…It was demonstrated that the subfield codes of ovoid codes are very attractive [11]. The parameters of some hyperoval codes and the conic codes were also studied in [14].…”

Some subfield codes from MDS codes

“…, A n ) is an important research object in coding theory, as it contains crucial information about the error correcting capability of the code. Thus study on weight distribution of linear codes attracts much attention in coding theory and many works focus on determining weight distribution of linear codes (see, for example, [6,7,8,9,10,11,15,21,22,23,24] and the references therein). A code C is said to be a t-weight code if the number of nonzero A i in the sequence (A 1 , A 2 , • • • , A n ) is equal to t. Denote by C ⊥ the dual code of a linear code C. We call an [n, k, d] code distance-optimal if no [n, k, d + 1] code exists and dimensionoptimal if no [n, k + 1, d] code exists.…”