This article addresses some foundational issues that arise in the study of linear codes defined over finite rings. Linear coding theory is particularly well-behaved over finite Frobenius rings. This follows from the fact that the... more
This article addresses some foundational issues that arise in the study of linear codes defined over finite rings. Linear coding theory is particularly well-behaved over finite Frobenius rings. This follows from the fact that the character module of a finite ring is free if and only if the ring is Frobenius.
So far we had a k-free (partial) recurrence for Eσ(n, k) (for σ = 2, 3) (that implied recurrences for Gσ(n)) and the trivial n-free (partial) recurrence, (5), for the general Eσ(n, k), It turns out that Eσ(n, k) satisfies a pure... more
So far we had a k-free (partial) recurrence for Eσ(n, k) (for σ = 2, 3) (that implied recurrences for Gσ(n)) and the trivial n-free (partial) recurrence, (5), for the general Eσ(n, k), It turns out that Eσ(n, k) satisfies a pure (ordinary) recurrence in k, and that Gσ(n) satisfies a linear recurrence with polynomial coefficients, in n, for every σ.
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With the help of some new results about weight enumerators of self-dual codes over <img src="/fulltext-image.asp?format=htmlnonpaginated&src=G88386122777T8U7_html\10801_2004_Article_123924_TeX2GIFIE2.gif" border="0"... more
With the help of some new results about weight enumerators of self-dual codes over <img src="/fulltext-image.asp?format=htmlnonpaginated&src=G88386122777T8U7_html\10801_2004_Article_123924_TeX2GIFIE2.gif" border="0" alt=" $$\mathbb{Z}_4 $$ " /> we investigate a class of double circulant codes over <img src="/fulltext-image.asp?format=htmlnonpaginated&src=G88386122777T8U7_html\10801_2004_Article_123924_TeX2GIFIE3.gif" border="0" alt=" $$\mathbb{Z}_4 $$ " />, one of which leads to an extremal even unimodular 40–dimensional lattice. It is conjectured that there should be “Nine more constructions
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This chapter describes algorithms which, given an arbitrary point of R n , find the closest point of some given lattice. The lattices discussed include the root lattices A n , D n , E 6 E 7, E 8 and their duals. These algorithms can be... more
This chapter describes algorithms which, given an arbitrary point of R n , find the closest point of some given lattice. The lattices discussed include the root lattices A n , D n , E 6 E 7, E 8 and their duals. These algorithms can be used for vector quantizing or for decoding lattice codes for a bandlimited channel.
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Contributions of new records, explicit codes or upper bounds are welcomed. All contributions will be acknowledged. ... A(n,d) >= N*2^(k-(n0-n)). ... The word exact in the Reference column indicates that this is the exact value of... more
Contributions of new records, explicit codes or upper bounds are welcomed. All contributions will be acknowledged. ... A(n,d) >= N*2^(k-(n0-n)). ... The word exact in the Reference column indicates that this is the exact value of A(n,d). So far this only appears for a few entries (although more are known to be exact). ... The table covers the range minimal distance d <= 29 and length n <= 512. ... The primary source for this table was Simon Litsyn's paper "Tables of Best Known Binary Codes", which appeared as Chapter 5, pages 463-498, in the Handbook of ...
We prove that the minimal length of a word $S_n$ having the property that it contains exactly $F_{m+2}$ distinct subwords of length $m$ for $1 \leq m \leq n$ is $F_n + F_{n+2}$. Here $F_n$ is the $n$th Fibonacci number defined by $F_1 =... more
We prove that the minimal length of a word $S_n$ having the property that it contains exactly $F_{m+2}$ distinct subwords of length $m$ for $1 \leq m \leq n$ is $F_n + F_{n+2}$. Here $F_n$ is the $n$th Fibonacci number defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. We also give an algorithm that generates a minimal word $S_n$ for each $n \geq 1$.
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... Journal of Human Hypertension; The Journal of Investigative Dermatology; JID SymposiumProceedings; Journal of Perinatology. ... Lipidomics Gateway; Nature Milestones; Nature Network;Nature News; Nature Precedings; Neuroscience... more
... Journal of Human Hypertension; The Journal of Investigative Dermatology; JID SymposiumProceedings; Journal of Perinatology. ... Lipidomics Gateway; Nature Milestones; Nature Network;Nature News; Nature Precedings; Neuroscience Gateway; Omics Gateway; Pathway Interaction ...
Research Interests: Biochemistry, Bioinformatics, Evolutionary Biology, Environmental Science, Developmental Biology, and 15 moreClimate Change, Computational Biology, Biotechnology, Cancer, Biology, Cell Cycle, Ecology, Drug Discovery, Evolution, Astrophysics, Functional Genomics, Astronomy, DNA, Cell Signalling, and Earth Science
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... complicated. Applications of theorem 5.1. 4 For the ternary Golay code (example 11 of section 1), V-ai2+ 5&amp;amp;quot;^ i2&amp;amp;#x27; For PLESS&amp;amp;#x27;s [24, 12, 9] symmetry code ([35],[36]), V= 179a2 19 R+... more
... complicated. Applications of theorem 5.1. 4 For the ternary Golay code (example 11 of section 1), V-ai2+ 5&amp;amp;quot;^ i2&amp;amp;#x27; For PLESS&amp;amp;#x27;s [24, 12, 9] symmetry code ([35],[36]), V= 179a2 19 R+ 595C2 _ 352 432 12^ 4* 12 12 432 12 9^ 24&amp;amp;#x27; Page 147. ...