Maximum Minimum Distance Equidistant (n, M, D; Q) Codes For Fixed N, M, Q

The construction of good codes is a basic problem in coding theory. It is well known that combinatorial design theory and coding theory are closely related. Certain combinatorial structures have been used to construct good codes such that the parameters of the codes achieve their optimality. A (n, M, d; q) code is called equidistant code if the Hamming distance between any two codewords is d, where the code length of the q-ary code is n and the code size is M. It was proved that for any equidistant (n, M, d; q) code, d≤((nM(q-1)))/(((M-1)q))(=d_opt, say). An equidistant (n, M, d; q) code that achieves this equality is said to be optimal. A necessary condition for the existence of an optimal equidistant code is that d_opt should be an integer. If d_opt is not an integer, then no optimal equidistant code exists, in this case, for fixed n, M, q, the code with d=「d_opt」is called a maximum minimum distance equidistant code, which is obviously the best possible one among equidistant codes with parameters n, M and q. We know that using minimum distance decoding, a code with minimum distance d can correct「(d-1)/2」errors and detect「d/2」errors. So discussing the maximum minimum distance of an equidistant code has important theoretical significance.In this dissertation, the construction of maximum minimum distance equidistant (n, M, d; q) codes for fixed n, M, q is investigated. We consider the problem of constructing maximum minimum distance equidistant codes from a combinatorial viewpoint and make use of the known results of certain combinatorial designs. This dissertation is divided into five chapters.In chapter one, we simply introduce the background about the development of equidistant code, and give some basic concepts and notions.In chapter two, at first, we present the equivalence between the equidistant code and the equidistant array, then we show the close relationship among balanced array, orthogonal array and equidistant array. After that, we construct equidistant codes from balanced arrays, and we obtain the corresponding maximum minimum distance equidistant codes. The following are the main results of this chapter.Theorem 2.2.1 Suppose 4t-1 is a prime power, then there exists an optimal equidistant (2(4t-1), 4t, 4t; 2)code.Theorem 2.2.2 If s and t are integers such that st+1 is a prime power, then there exists an equidistant (s(st+1), st+1, s(st-t+2); s+1) code.Corollary 2.2.1 If s is an integer such that 2s+1 is a prime power, then there exists a maximum minimum distance equidistant (s(2s+1), 2s+1, 2s²; s+1) code.In chapter three, we use SBIBD to construct equidistant code, and we also obtain maximum minimum distance equidistant codes. The main results are listed as the following.Lemma 3.2.1 If there exists a (v, k,λ)-SBIBD, then there exists an equidistant (m, v, d; 4) code, where m=(₂^v), d=(₂^v)-((₂^λ)+λ(k-λ)+(₂^(v-2k+λ)).Theorem 3.2.1 Suppose 4t-1 is a prime power, then there exists a maximum minimum distance equidistant ((₂^4t-1)), 4t-1, 3t(2t-1); 4) code.Theorem 3.2.2 Suppose 4t-1 is a prime power, then there exists an optimal equidistant ((₂^4t-1)), 4t, 3t(2t-1); 4)code.In chapter four, we prove that nested balanced incomplete block designs can also give equidistant codes, and we construct some codes by first constructing nested BIB design from the known results of mutually orthogonal Latin squares. Furthermore, we obtain some relevant maximum minimum distance equidistant codes. The following are the main results of chapter four.Theorem 4.1.1 If there exists a (v; k₁,λ₁; k₂,λ₂) nested BIB design, then there exists an equidistant (b₁, v, 2r-λ₁-λ₂; 1+k₁/k₂) code.Corollary 4.2.1 For any prime power q, for any integer k₁≤q, and for any integer k₂＞1 such that k₂ divides k₁, there exists a (q; k₁, k₁(k₁-1); k₂, k₁(k₂-1)) nested BIB design.Corollary 4.2.2 For any odd prime power q, for any odd integer k₁≤q, and for any integer k₂＞1 such that k₂ divides k₁, there exists a (q; k₁, k₁(k₁-1)/2; k₂, k₁(k₂-1)/2) nested BIB design.Theorem 4.3.1 Let q be a prime power, k₁≤q an integer, k₂＞1 an integer such that k₂ divides k₁. If |q-(k₁+k₂)|＜((1+(k₂)/(k₁))^1/2, then there exists a maximum minimum distance equidistant (q(q-1), q, 2k₁q-k₁(k₁+k₂); 1+(k₁)/(k₂)) code.Theorem 4.3.2 Let q be an odd prime power, k₁≤q an odd integer, k₂＞1 an integer such that k₂ divides k₁. If |q-(k₁+k₂)|＜(2(1+(k₂)/(k₁))^1/2, then there exists a maximum minimum distance equidistant ((q(q-1))/2, q, k₁q-(k₁(k₁+k₂))/2; 1+(k₁)/(k₂))code.In the last part of this dissertation, some problems for further research are presented.