# American Institute of Mathematical Sciences

• Previous Article
Further results on implicit factoring in polynomial time
• AMC Home
• This Issue
• Next Article
Infinite families of recursive formulas generating power moments of ternary Kloosterman sums with square arguments arising from symplectic groups
May  2009, 3(2): 179-203. doi: 10.3934/amc.2009.3.179

## On isometries for convolutional codes

 1 Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027

Received  February 2009 Revised  April 2009 Published  May 2009

In this paper we will discuss isometries and strong isometries for convolutional codes. Isometries are weight-preserving module isomorphisms whereas strong isometries are, in addition, degree-preserving. Special cases of these maps are certain types of monomial transformations. We will show a form of MacWilliams Equivalence Theorem, that is, each isometry between convolutional codes is given by a monomial transformation. Examples show that strong isometries cannot be characterized this way, but special attention paid to the weight adjacency matrices allows for further descriptions. Various distance parameters appearing in the literature on convolutional codes will be discussed as well.
Citation: Heide Gluesing-Luerssen. On isometries for convolutional codes. Advances in Mathematics of Communications, 2009, 3 (2) : 179-203. doi: 10.3934/amc.2009.3.179
 [1] Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415 [2] Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Advances in Mathematics of Communications, 2010, 4 (1) : 69-81. doi: 10.3934/amc.2010.4.69 [3] David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131 [4] Liqun Qi, Chen Ling, Jinjie Liu, Chen Ouyang. An orthogonal equivalence theorem for third order tensors. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021154 [5] Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543 [6] Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55 [7] David Keyes. $\mathbb F_p$-codes, theta functions and the Hamming weight MacWilliams identity. Advances in Mathematics of Communications, 2012, 6 (4) : 401-418. doi: 10.3934/amc.2012.6.401 [8] José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Convolutional codes with a matrix-algebra word-ambient. Advances in Mathematics of Communications, 2016, 10 (1) : 29-43. doi: 10.3934/amc.2016.10.29 [9] Keonhee Lee, Kazuhiro Sakai. Various shadowing properties and their equivalence. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 533-540. doi: 10.3934/dcds.2005.13.533 [10] Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161 [11] Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221 [12] Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83 [13] José Ignacio Iglesias Curto. Generalized AG convolutional codes. Advances in Mathematics of Communications, 2009, 3 (4) : 317-328. doi: 10.3934/amc.2009.3.317 [14] Diego Napp, Carmen Perea, Raquel Pinto. Input-state-output representations and constructions of finite support 2D convolutional codes. Advances in Mathematics of Communications, 2010, 4 (4) : 533-545. doi: 10.3934/amc.2010.4.533 [15] Michael C. Sullivan. Invariants of twist-wise flow equivalence. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 475-484. doi: 10.3934/dcds.1998.4.475 [16] Giuseppe Buttazzo, Luigi De Pascale, Ilaria Fragalà. Topological equivalence of some variational problems involving distances. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 247-258. doi: 10.3934/dcds.2001.7.247 [17] Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047 [18] Zemer Kosloff, Terry Soo. The orbital equivalence of Bernoulli actions and their Sinai factors. Journal of Modern Dynamics, 2021, 17: 145-182. doi: 10.3934/jmd.2021005 [19] Mike Crampin, David Saunders. Homogeneity and projective equivalence of differential equation fields. Journal of Geometric Mechanics, 2012, 4 (1) : 27-47. doi: 10.3934/jgm.2012.4.27 [20] Mrinal Kanti Roychowdhury, Daniel J. Rudolph. Nearly continuous Kakutani equivalence of adding machines. Journal of Modern Dynamics, 2009, 3 (1) : 103-119. doi: 10.3934/jmd.2009.3.103

2020 Impact Factor: 0.935