Group codes for the Gaussian channel (Abstr.)

A class of equal-energy codes for use on the Gaussian channel is defined and investigated. Members of the class are eared group codes because of the manner in which they can be generated from a group of orthogonal matrices. Group codes possess an important symmetry property. Roughly speaking, all words in such a code are on an equal footing: each has the same error probability (under the assumptions of the usual model) and each has the same disposition of neighbors. A number of theorems about such codes are proved. A decomposition theorem shows every group code to be equivalent to a direct sum of certain basic group codes generated by real-irreducible representations of a finite group associated with the code. Some theorems on distances between words in group codes are demonstrated. The difficult problem of finding group codes with large nearest neighbor distance is discussed in detail and formulated in several ways. It is noted that linear (or group) codes for the binary channel can be regarded as very speciM cases of the group codes discussed. A definition of a group code for the Gaussian channel follows. An equal-energy code <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> with parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> for this channel is a collection of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> distinct unit <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -vectors, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X₁, X₂, ⋯ , XM</tex> say, that span a Euclidean <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -space. An <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n × n</tex> orthogonal matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</tex> is said to be a symmetry of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> if the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> vectors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Yᵢ = 0Xᵢ, i = 1, 2, ⋯ , M</tex> are again the collection <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> . The set of all symmetries of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> , say <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0₁, 0₂, ⋯ , 0g</tex> , forms a group <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{G}(C)</tex> under matrix multiplication. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{G}(C)</tex> contains <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> elements <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0_{α₁}, 0_{α₂}, ⋯ , 0α M</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Xᵢ = 0α iX₁, i = 1, 2, ⋯ , M</tex> , then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> is called a group code.