Abstract
It has been shown that maximum distance profile (MDP) convolutional codes have optimal recovery rate for windows of a certain length, when transmitting over an erasure channel. In addition, the subclass of complete MDP convolutional codes has the ability to reduce the waiting time during decoding. Since so far general constructions of these codes have only been provided over fields of very large size, there arises the question about the necessary field size such that these codes could exist. In this paper, we derive upper bounds on the necessary field size for the existence of MDP and complete MDP convolutional codes and show that these bounds improve the already existing ones. For some special choices of the code parameters, we are even able to give the exact minimum field size. Moreover, we derive lower bounds for the probability that a random code is MDP respective complete MDP.
Similar content being viewed by others
References
Almeida P.J., Napp D., Pinto R.: Superregular matrices and applications to convolutional codes. Linear Algebra Appl. 499, 1–25 (2016).
Almeida P.J., Napp D., Pinto R.: A new class of superregular matrices and MDP convolutional codes. Linear Algebra Appl. 439, 2145–2157 (2013).
Barbero A., Ytrehus O.: Rate \((n-1)/n\) systematic memory maximum distance separable convolutional codes. IEEE Trans. Inf. Theory 64(4), 3018–3030 (2018).
Gluesing-Luerssen H., Rosenthal J., Smarandache R.: Strongly-MDS convolutional codes. IEEE Trans. Inf. Theory 52(2), 584–598 (2006).
Hutchinson R.: The existence of strongly MDS convolutional codes. SIAM J. Control Optim. 47, 2812–2826 (2008).
Hutchinson R., Rosenthal J., Smarandache R.: Convolutional codes with maximum distance profile. Syst. Control Lett. 54, 53–63 (2005).
Hutchinson R., Smarandache R., Trumpf J.: On superregular matrices and MDP convolutional codes. Linear Algebra Appl. 428, 2585–2596 (2008).
Lieb J.: Complete MDP convolutional codes. J. Algebra Appl. arXiv:1712.08767 (2018).
Lieb J.: Counting Polynomial Matrices over Finite Fields. Matrices with Certain Primeness Properties and Applications to Linear Systems and Coding Theory (Dissertation), Wuerzburg University Press. (2017). https://www.bod.de/buchshop/couting-polynomial-matrices-over-finite-fields-julia-lieb-9783958260641.
Lieb J.: The probability of primeness for specially structured polynomial matrices over finite fields with applications to linear systems and convolutional codes. Math. Control Signals Syst. 29, 8 (2017). https://doi.org/10.1007/s00498-017-0191-z.
Napp D., Smarandache R.: Constructing strongly MDS convolutional codes with maximum distance profile. Adv. Math. Commun. 10(2), 275–290 (2016).
Rosenthal J.: Connections between linear systems and convolutional codes. In: Marcus B., Rosenthal J. (eds.) Codes, Systems and Graphical Models IMA, vol. 123, pp. 39–66 (2001).
Rosenthal J., Smarandache R.: Maximum distance separable convolutional codes. Appl. Algebra Eng. Commun. Comput. 10, 15–32 (1999).
Schwartz J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701–717 (1980).
Tomas V., Rosenthal J., Smarandache R.: Decoding of convolutional codes over the erasure channel. IEEE Trans. Inf. Theory 58(1), 90–108 (2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Panario.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lieb, J. Necessary field size and probability for MDP and complete MDP convolutional codes. Des. Codes Cryptogr. 87, 3019–3043 (2019). https://doi.org/10.1007/s10623-019-00661-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00661-6