# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

2007 , Volume 1 , Issue 1

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2007, 1(1): i-ii doi: 10.3934/amc.2007.1.1i +[Abstract](111) +[PDF](53.2KB)
Abstract:
Almost 60 years have passed since Claude E.~Shannon's celebrated two-part article A Mathematical Theory of Communication appeared in July and October of 1948. With that paper, Shannon founded a discipline, whose wide reach could hardly have been foreseen at the time.
To commemorate the occasion of the 50th anniversary of the birth of Information Theory as a discipline, the IEEE Transactions on Information Theory published a special issue that included the following statement in its preface:
With communication engineering at the epicenter of the bombshell, the sensational aftermath of Shannon's paper soon reached Mathematics, Physics, Statistics, Computing, and Cryptology. Even Economics, Biology, Linguistics and other fields in the natural and social sciences felt the ripples of Shannon's new theory.

2007, 1(1): 1-11 doi: 10.3934/amc.2007.1.1 +[Abstract](131) +[PDF](147.1KB)
Abstract:
The paper analyzes CFVZ, a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system is dominated by the complexity of solving a fixed number of discrete logarithm problems in the group of an elliptic curve. Using an adapted Pollard rho algorithm it is shown that this problem is essentially as hard as solving one discrete logarithm problem in the group of an elliptic curve. Hence, the CFVZ cryptosystem has no advantages over traditional elliptic curve cryptography and should not be used in practice.
2007, 1(1): 13-28 doi: 10.3934/amc.2007.1.13 +[Abstract](146) +[PDF](223.4KB)
Abstract:
Starting with Shoup's seminal paper [24], the generic group model has been an important tool in reductionist security arguments. After an informal explanation of this model and Shoup's theorem, we discuss the danger of flaws in proofs. We next describe an ontological difference between the generic group assumption and the random oracle model for hash unctions. We then examine some criticisms that have been leveled at the generic group model and raise some questions of our own.
2007, 1(1): 29-44 doi: 10.3934/amc.2007.1.29 +[Abstract](196) +[PDF](211.7KB)
Abstract:
An anonymous membership broadcast scheme is a method in which a sender broadcasts the secret identity of one out of a set of $n$ receivers, in such a way that only the right receiver knows that he is the intended receiver, while the others can not determine any information about this identity (except that they know that they are not the intended ones). In a $w$-anonymous membership broadcast scheme no coalition of up to $w$ receivers, not containing the selected receiver, is able to determine any information about the identity of the selected receiver. We present two new constructions of $w$-anonymous membership broadcast schemes. The first construction is based on error-correcting codes and we show that there exist schemes that allow a flexible choice of $w$ while keeping the plexities for broadcast communication, user storage and required randomness polynomial in log $n$. The second construction is based on the concept of collision-free arrays, which is introduced in this paper. The construction results in more flexible schemes, allowing trade-offs between different complexities.
2007, 1(1): 45-64 doi: 10.3934/amc.2007.1.45 +[Abstract](162) +[PDF](233.0KB)
Abstract:
Two class association schemes consist of either strongly regular graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary codes attached to rank three groups.
2007, 1(1): 65-81 doi: 10.3934/amc.2007.1.65 +[Abstract](129) +[PDF](246.3KB)
Abstract:
A $(k, n)$-blocking multiset in the projective Hjelmslev plane PHG($R^3_R$) is defined as a multiset $\mathfrak K$ with $\mathfrak K(\mathcal P) = k$, $\mathfrak K(l) \geq n$ for any line $l$ and $\mathfrak K(l_0) = n$ for at least one line $l_0$. In this paper, we investigate blocking sets in projective Hjelmslev planes over chain rings $R$ with $|R| = q^m, R$∕rad$R \cong \mathbb F_q, q = p^r, p$ prime. We prove that for a $(k, n)$-blocking multiset with $1 \leq n \leq q, k \geq n$qm-1$(q+1)$. The image of a $(n$qm-1$(q +1), n)$-blocking multiset with $n <$char$R$ under the the canonical map $\pi^{(1)}$ is ''sum of lines''. In particular, the smallest $(k, 1)$-blocking set is the characteristic function of a line and its cardinality is $k =$qm-1$(q + 1)$. We prove that if $R$ has a subring $S$ with $\sqrt R$ elements that is a chain ring such that $R$ is free over $S$ then the subplane PHG($S^3_S$) is an irreducible $1$-blocking set in PHG($R^3_R$). Corollaries are derived for chain rings with $|R| = q^2, R$∕rad$R \cong \mathbb F_q$.
In case of chain rings $R$ with $|R| = q^2, R$∕rad$R \cong \mathbb F_q$ and $n = 1$, we prove that the size of the second smallest irreducible $(k, 1)$-blocking set is $q^2 + q + 1$. We classify all blocking sets with this cardinality. It turns out that if char$R = p$ there exist (up to isomorphism) two such sets; if char$R = p^2$ the irreducible $(q^2 + q + 1, 1)$-blocking set is unique. We introduce a class of irreducible $(q^2 + q + s, 1)$ blocking sets for every $s \in {1, \cdots , q + 1}$. Finally, we discuss briefly the codes over $\mathbb F_q$ obtained from certain blocking sets.
2007, 1(1): 83-92 doi: 10.3934/amc.2007.1.83 +[Abstract](99) +[PDF](160.6KB)
Abstract:
New bounds on the rate of non-binary codes and non-binary constant weight codes are derived. The asymptotic forms of these bounds outperform known bounds in a wide range of distances. The method is based on analysis of subsets in products of Hamming and Johnson association schemes.
2007, 1(1): 93-97 doi: 10.3934/amc.2007.1.93 +[Abstract](161) +[PDF](118.4KB)
Abstract:
We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems. Specifically, we prove that if almost all codes (in the class of linear or nonlinear codes) are good packings, then only a vanishing fraction of codes are good coverings, and vice versa: if almost all codes are good coverings, then at most a vanishing fraction of codes are good packings. We also show that any specific maximal binary code is either a good packing or a good covering, in a certain well-defined sense.
2007, 1(1): 99-109 doi: 10.3934/amc.2007.1.99 +[Abstract](138) +[PDF](161.7KB)
Abstract:
We obtain finite and asymptotic Gilbert-Varshamov type bounds for linear codes over finite chain rings with various weights.
2007, 1(1): 111-130 doi: 10.3934/amc.2007.1.111 +[Abstract](157) +[PDF](229.8KB)
Abstract:
Binary self-dual codes and additive self-dual codes over $\mathbb F_4$ have in common interesting properties, for example, Type I, Type II, shadows, etc. Recently Bachoc and Gaborit introduced the notion of $s$-extremality for binary self-dual codes, generalizing Elkies' study on the highest possible minimum weight of the shadows of binary self-dual codes. In this paper, we introduce a concept of $s$-extremality for additive self-dual codes over $\mathbb F_4$, give a bound on the length of these codes with even distance $d$, classify them up to minimum distance $d = 4$, give possible lengths and (shadow) weight enumerators for which there exist $s$-extremal codes with $5 \leq d \leq 11$ and give five $s$-extremal codes with $d = 7$. We construct four $s$-extremal codes of length $n = 13$ and minimum distance $d = 5$. We relate an $s$-extremal code of length $3d$ to another $s$-extremal code of that length, and produce extremal Type II codes from $s$-extremal codes.
2007, 1(1): 131-149 doi: 10.3934/amc.2007.1.131 +[Abstract](104) +[PDF](255.8KB)
Abstract:
We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps. Several new upper bounds on codes in caps are derived. Applications of these bounds to estimates of the kissing numbers and one-sided kissing numbers are considered.
It is proved that the maximum size of codes in spherical caps for large dimensions is determined by the maximum size of spherical codes, so these problems are asymptotically equivalent.
2007, 1(1): 151-171 doi: 10.3934/amc.2007.1.151 +[Abstract](108) +[PDF](270.6KB)
Abstract:
Order domains are a class of commutative rings introduced by Høholdt, van Lint, and Pellikaan to simplify the theory of error control codes using ideas from algebraic geometry. The definition is largely motivated by the structures utilized in the Berlekamp-Massey-Sakata (BMS) decoding algorithm, with Feng-Rao majority voting for unknown syndromes, applied to one-point geometric Goppa codes constructed from curves. However, order domains are much more general, and O'Sullivan has shown that the BMS algorithm can be used to decode codes constructed from order domains by a suitable generalization of Goppa's construction for curves. In this article we will first discuss the connection between order domains and valuations on function fields over a finite field. Under some mild conditions, we will see that a general projective variety over a finite field has projective models which can be used to construct order domains and Goppa-type codes for which the BMS algorithm is applicable. We will then give a slightly different interpretation of Geil and Pellikaan's extrinsic characterization of order domains via the theory of Gröbner bases, and show that their results are related to the existence of toric deformations of varieties. To illustrate the potential usefulness of these observations, we present a series of new explicit examples of order domains associated to varieties with many rational points over finite fields: Hermitian hypersurfaces, Deligne-Lusztig varieties, Grassmannians, and flag varieties.

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