Codes of small defect from finite geometries: embeddings, bounds, and constructions.

所提出的研究涉及有限几何产生的纠错码。我们将利用有限射影空间的子结构来研究良码的构造。(n，k，d)_q-码是大小为q的字母表上的q^k个n元组(码字)的集合，使得没有两个码字在n-d坐标下一致。如果码是向量空间，则码是线性的。C的单态亏损为S(C)=n-k+1-d。编码论中的一个基本问题是：问题1：对于固定的k，q和S(C)，确定n的最大值。对于小缺陷码，我们将与问题1发生冲突。S(C)=0的码称为最大距离可分(MDS码)。射影空间中的线性MDS码和弧是等价的对象。MDS码具有最强的纠错能力，从深空通信到光盘记录都有许多应用。然而，所有已知的线性MDS码都必然是短的，许多应用将受益于较长的码。这就引出了一个问题，即是否可以找到更长的(非线性)MDS码。其次是S(C)=1的AMDS(几乎-MDS)码。没有重复坐标的线性AMDS码称为NMDS(NMDS)。线性(n，k，d)_q-NMDS码和PG(k，q)中的(n，k+1)-弧是等价对象。三次曲线的有理点提供(n，k+1)-圆弧。与MDS码不同，AMDS码超过曲线提供的界限(长度)的例子很多。我们将研究问题1以及相关的MDS和AMDS码的扩展、嵌入和唯一性问题。

The proposed research concerns error correcting codes arising from finite geometries. We shall investigate constructions of good codes using substructures of finite projective spaces.  An (n,k,d)_q-code is a collection of q^k n-tuples (codewords) over an alphabet of size q such that no two codewords agree in as many as n-d coordinates. A code is linear if it is a vector space. The singleton defect of C is S(C)=n-k+1-d. A fundamental problem in coding theory is the following:Problem 1: Determine the maximum value of n for fixed k, q, and S(C).We shall skirmish with Problem 1 for codes of small defect.Codes with S(C)=0 are called MDS (Maximum Distance Separable) codes.  Linear MDS codes and arcs in projective spaces are equivalent objects. MDS codes possess the strongest possible error correction ability and have found many applications from deep space communications to compact disk recording.  However, all known linear MDS codes are necessarily short and many applications would benefit from longer codes. This begs the question as to whether longer (nonlinear) MDS codes may be found.The next best codes are those with S(C)=1, the AMDS (Almost-MDS) codes. A linear AMDS code without repeated coordinates is Near-MDS (NMDS). Linear (n,k,d)_q-NMDS codes and (n,k+1)-arcs in PG(k,q) are equivalent objects. Rational points of cubic curves provide (n,k+1)-arcs. Unlike MDS codes, there are many examples of AMDS codes exceeding the bounds (on length) provided by curves. We shall investigate Problem 1 and related extending, embedding, and uniqueness problems for MDS and AMDS codes.