One of the most remarkable and beautiful theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes and their connections with invariant theory. In the past ...35 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes. This self-contained book develops a new theory which is powerful enough to include all the earlier generalizations. It is also in part an encyclopedia that gives a very extensive list of the different types of self-dual codes and their properties, including tables of the best codes that are presently known. Besides self-dual codes, the book also discusses two closely-related subjects, lattices and modular forms, and quantum error-correcting codes. This book, written by the leading experts in the subject, has no equivalent in the literature and will be of great interest to mathematicians, communication theorists, computer scientists and physicists.
A coloring‐book approach to finding coordination sequences Goodman-Strauss, C.; Sloane, N. J. A.
Acta crystallographica. Section A, Foundations and advances,
January 2019, 2019-Jan-01, 2019-01-01, 20190101, Volume:
75, Issue:
1
Journal Article
Peer reviewed
Open access
An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the two kinds of vertices (tetravalent and ...trivalent) in the Cairo (or dual‐32.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8, 12, 16, …, the same as for a vertex in the familiar square (or 44) tiling. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used to obtain coordination sequences for the 32.4.3.4, 3.4.6.4, 4.82, 3.122 and 34.6 uniform tilings, and the snub‐632 tiling. In several cases the results provide proofs for previously conjectured formulas.
This article presents a simple method for finding formulas for coordination sequences, based on coloring the underlying graph according to certain rules. It is illustrated by applying it to several uniform tilings and their duals.
We present a novel technique for encoding and decoding constant weight binary vectors that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a ...Euclidean space of dimension equal to the weight of the code. The encoder and decoder mappings are then interpreted as a bijection between a certain hyper-rectangle and a polytope in this Euclidean space. An inductive dissection algorithm is developed for constructing such a bijection. We prove that the algorithm is correct and then analyze its complexity. The complexity depends on the weight of the vector, rather than on the block length as in other algorithms. This approach is advantageous when the weight is smaller than the square root of the block length.
Following an initiative of the late Hans Zassenhaus in 1965, the Departments of Mathematics at The Ohio State University and Denison University organize conferences in combinatorics, group theory, ...and ring theory. Between May 18-21, 2000, the 25th conference of this series was held. Usually, there are twenty to thirty invited 20-minute talks in each of the three main areas. However, at the 2000 meeting, the combinatorics part of the conference was extended, to honor the 65th birthday of Professor Dijen Ray-Chaudhuri. This volulme is the proceedings of this extension. Most of the papers are in coding theory and design theory, reflecting the major interest of Professor Ray-Chaudhuri, but there are articles on association schemes, algebraic graph theory, combinatorial geometry, and network flows as well. There are four surveys and seventeen research articles, and all of these went through a thorough refereeing process. The volume is primarily recommended for researchers and graduate students interested in new developments in coding theory and design theory.
Until 1973 there was no database of integer sequences. Someone coming across the sequence 1, 2, 4, 9, 21, 51, 127,... would have had no way of discovering that it had been studied since 1870 (today ...these are called the Motzkin numbers, and form entry A001006 in the database). Everything changed in 1973 with the publication of "A Handbook of Integer Sequences", which listed 2372 entries. This report describes the fifty-year evolution of the database from the "Handbook" to its present form as "The On-Line Encyclopedia of Integer Sequences" (or OEIS), which contains 360,000 entries, receives a million visits a day, and has been cited 10,000 times, often with a comment saying "discovered thanks to the OEIS".
A Note on Projecting the Cubic Lattice Sloane, N. J. A.; Vaishampayan, Vinay A.; Costa, Sueli I. R.
Discrete & computational geometry,
10/2011, Volume:
46, Issue:
3
Journal Article
Peer reviewed
Open access
It is shown that, given any (
n
−1)-dimensional lattice Λ, there is a vector
v
∈ℤ
n
such that the orthogonal projection of ℤ
n
onto
v
⊥
is, up to a similarity, arbitrarily close to Λ. The problem ...arises in attempting to find the largest cylinder anchored at two points of ℤ
n
and containing no other points of ℤ
n
.
What is the smallest number of pieces that you can cut an n-sided regular polygon into so that the pieces can be rearranged to form a rectangle? Call it r(n). The rectangle may have any proportions ...you wish, as long as it is a rectangle. The rules are the same as for the classical problem where the rearranged pieces must form a square. Let s(n) denote the minimum number of pieces for that problem. For both problems the pieces may be turned over and the cuts must be simple curves. The conjectured values of s(n), 3 <= n <= 12, are 4, 1, 6, 5, 7, 5, 9, 7, 10, 6. However, only s(4)=1 is known for certain. The problem of finding r(n) has received less attention. In this paper we give constructions showing that r(n) for 3 <= n <= 12 is at most 2, 1, 4, 3, 5, 4, 7, 4, 9, 5, improving on the bounds for s(n) in every case except n=4. For the 10-gon our construction uses three fewer pieces than the bound for s(10). Only r(3) and r(4) are known for certain. We also briefly discuss q(n), the minimum number of pieces needed to dissect a regular n-gon into a monotile.
The ``comma sequence'' starts with 1 and is defined by the property that if k and k' are consecutive terms, the two-digit number formed from the last digit of k and the first digit of k' is equal to ...the difference k'-k. If there is more than one such k', choose the smallest, but if there is no such k' the sequence terminates. The sequence begins 1, 12, 35, 94, 135, ... and, surprisingly, ends at term 2137453, which is 99999945. The paper analyzes the sequence and its generalizations to other starting values and other bases. A slight change in the rules allows infinitely long comma sequences to exist.
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