Abstract
In this work, the correspondence between linear (n,k,d) codes and aperiodic convolution algorithms for computing a systemφ ofk bilinear forms over GF(pm) is explored. A number of properties are established for the linear codes that can be obtained from a computational procedure of this type. A particular bilinear form is considered and a class of linear codes over GF(2m) is derived with varyingk andd parameters. The code lengthn is equal to the multiplicative complexity of the computation of an aperiodic convolution and an efficient computation thereof leads to the shortest codes possible using this approach, many of which are optimal or near-optimal. A new decoding procedure for this class of linear codes is presented which exploits the block structure of the generator matrix of the codes. Several interesting observations are made on the nature of the codes obtained as a result of such computations. Such a computation of bilinear forms can be generalized to include other bilinear forms and the related classes of codes.
Similar content being viewed by others
References
Aho, A. V., Hopcroft, J. E., Ullman, J. D.: The design and analysis of computer algorithms. London: Addison-Wesley 1974
Brockett, R. W., Dobkin, D.: On the optimal evolution of a set of bilinear forms. Proceedings Fifth Annual ACM Symposium on Theory of Computing, pp. 88–95 (1973)
Dornstetter, J. L.: On the computation of the product of two polynomials over a finite field. Int. Symp. on Info. Theory 1983
Helgert, H. J., Stinaff, R. D.: Minimum-distance bounds for binary linear codes. IEEE Trans. Information TheoryIT-19, 344–356 (1973)
Hopcroft, J., Musinki, J.: Duality applied to the complexity of matrix multiplication and other bilinear forms. SIAM J. Computing2, 159–173 (1973)
Krishna, H., Morgera, S. D.: A new error control scheme for hybrid ARQ systems. IEEE Trans. Communications,COM-35, 981–990 (1987)
Lempel, A., Winograd, S.: A new approach to error-correcting codes. IEEE Trans. Information Theory.IT-23, 503–508 (1977)
MacWilliams, F. J., Sloane, N. J. A.: The theory of error-correcting codes. Amsterdam, New York: North Holland 1977
McClellan, J. H., Rader, C. M.: Number theory in digital signal processing. Englewood Cliffs: Prentice-Hall 1979
Wagh, M. D., Morgera, S. D.: A new structured design method for convolutions over finite fields, Part I. IEEE Trans. Information Theory,IT-29(4), 583–595 (1983)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Krishna, H. On a signal processing algorithms based class of linear codes. AAECC 4, 41–57 (1993). https://doi.org/10.1007/BF01270399
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01270399