MacWilliams extension theorems and the local–global property for codes over Frobenius rings

MacWilliams Extension Theorems and the Local-Global Property for Codes over Rings

The MacWilliams extension theorem is investigated for various weight functions over finite Frobenius rings. The problem is reformulated in terms of a local-global property for subgroups of the general linear group. Among other things, it is shown that the extension theorem holds true for poset weights if and only if the underlying poset is hierarchical. Specifically, the Rosenbloom-Tsfasman wei...

Witt Theorems for Linear Codes over Finite Frobenius Rings

Extension Theorems for Linear Codes over Finite Rings

Various forms of the extension problem are discussed for linear codes de ned over nite rings. The extension theorem for symmetrized weight compositions over nite Frobenius rings is proved. As a consequence, an extension theorem for weight functions over certain nite commutative rings is also proved. The proofs make use of the linear independence of characters as well as the linear independence ...

Self-dual codes over commutative Frobenius rings

We prove that self-dual codes exist over all finite commutative Frobenius rings, via their decomposition by the Chinese Remainder Theorem into local rings. We construct non-free self-dual codes under some conditions, using self-dual codes over finite fields, and we also construct free self-dual codes by lifting elements from the base finite field. We generalize the building-up construction for ...

Extension theorems for self-dual codes over rings and new binary self-dual codes

In this work, extension theorems are generalized to self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F2m + uF2m for m = 1, 2. The duality and distance preserving Gray maps from F4 + uF4 to (F2 + uF2) and F42 are used to obtain self-dual codes whose binary Gray images are [64, 32, 12]-extremal self-dual. An F2 + uF2-exten...