[seqfan] Re: binary linear codes and orthogonal lattices

Neil Sloane njasloane at gmail.com
Tue Nov 28 04:50:58 CET 2017


Andrey,  Good catch.  Yes, I think they are the same, and I will do some
editing tomorrow.
However, the result follows from the theorem on page 64 of
Schwarzenberger's book, not from the thesis you mentioned.

Actually, the whole definition of "orthogonal lattice" is somewhat
unsatisfactory.

Gandikota defines it to be any lattice which is a rotated copy of Z^n.  If
we use
that definition then there is only one such lattice - it is unique, modulo
the action of O(n). Or there are infinitely many, if you count rotations as
different.
She is concerned with the difficulty of testing if a real n-D lattice is
orthogonal
- and she restricts herself to Construction A lattices (a term I introduced
in 1971, by the way,
although she gives the wrong reference)

Schwarzenberger's definition is different.  The theorem I mentioned
shows that whatever his definition is, the lattices are in one-to-one
correspondence with binary linear codes of length n containing no codewords
of weight 1.

By taking duals, this is equivalent to counting
binary linear codes with no coordinate identically zero, and that is what
A034343 enumerates.

So they are the same.

As for the conjecture "Conjecture (V. Kotesovec, 2010): a(n) = A178717(n-1)
- 1"
this never looked like more than an accident, and now we see that it is
false.



Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Mon, Nov 27, 2017 at 3:15 PM, Андрей Заболотский via SeqFan <
seqfan at list.seqfan.eu> wrote:

> Dear Seqfans,
>
> The known terms of  A007669 "Number of orthogonal lattices in dimension n"
> match the first terms of  A034343 "Number of inequivalent binary linear
> codes of length n and any dimension k <= n containing no column of zeros".
> Could they actually be the same sequence?
> Theorem 4.1.1 of a dissertation  https://www.cs.purdue.edu/
> homes/vgandiko/papers/thesis.pdf  states some correspondence between some
> orthogonal lattices and binary linear codes, but I'm not sure if it is
> relevant.
>
>
> Andrey Zabolotskiy
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>



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