Classification of convolutional codes
From their introduction by \textit{P. Elias} [IRE International Convention Record (part 4), 37--46 (1955)], convolutional codes attracted the interest of the error-correcting codes community due to their interesting properties, but at the beginning the study of these codes was largely heuristic. The interpretation of convolutional codes defined over a finite field \(\mathbb{F}\)\, as \(\mathbb{F}(z)\)-modules allowed an algebraic approach to them. In the present paper the authors apply methods from Algebraic Geometry to the classification of convolutional codes, constructing a moduli space for them. Section 2 recalls the concepts of block and convolutional codes and introduces the algebraic geometric tools necessary for the desired classification (coherent sheaves, Hilbert polynomial, quotient scheme). Section 3 explains the Kronecker-Hermite and the modified Kronecker-Hermite canonical generator matrices of a convolutional code and reproduces a result from \textit{P. A. Fuhrmann} and \textit{U. Helmke} [Linear Algebra Appl. (332-334), 265--353 (2001; Zbl 1031.93062)] giving a method for obtaining the modified Kronecker-Hermite canonical matrix. Section 4 achieves the wanted classification showing (Theorem 4.13) that the set of convolutional codes over \(\mathbb{F}\)\, with the same parameters (length, dimension, degree and memory) can be identified with the points of an algebraic subvariety of a Grassmannian. The last two sections show some applications of the classification. Section 5 proves three upper bounds for convolutional codes generalizing the classical bounds of Heller, Singleton and Griesmer, while Section 6 exploits the given representation of convolutional codes as points (which are block codes) in the Grassmannian as a way to derive convolutional codes with optimal free distance, showing the example of the two Hammning codes \(H_2(3)\)\, and \(H_3(3)\)\, as well as a Reed-Solomon code.
scientific article; zbMATH DE number 5689433
scientific article