The covering radius r of a code is the maximum distance from any vector in the space containing the code to the nearest codeword. The authors introduce a new function l(m,r), called the length function, which equals the smallest length of a binary code of codimension m and covering radius r. They investigate basic properties of the length function. Projective geometries over larger fields are used to construct families of codes which improve significantly the upper bound for l(m,2) obtained by amalgamation of Hamming codes. General methods are developed for ruling out the existence of codes of covering radius 2 with a given codimension and length resulting in lower bounds for l(m,2). A table is presented which gives the best results now known for l(m,r) with m>or=12 and r>or=12.
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机译：代码的覆盖半径r是从包含代码的空间中的任何矢量到最近的代码字的最大距离。作者介绍了一个新的函数l（m，r），称为长度函数，它等于余维m和覆盖半径r的二进制代码的最小长度。他们研究长度函数的基本属性。大域上的投影几何用于构造代码族，从而显着提高通过合并汉明码获得的l（m，2）的上限。已开发出通用方法来排除以给定的维数和长度覆盖半径2的代码的存在，从而导致l（m，2）的下限。给出了一个表格，给出了l（m，r）已知的最佳结果，其中m> or = 12且r> or = 12。
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