Semidefinite bounds for mixed binary/ternary codes

For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least $d$. For a nonnegative integer $k$, let $\mathcal{C}_k$ denote the collection of codes of cardinality at most $k$. For $D \in \mathcal{C}_k$, define $S(D) := \{C \in \mathcal{C}_k \mid D \subseteq C, |D| +2|C\setminus D| \leq k\}$. Then $N(n_2,n_3,d)$ is upper bounded by the maximum value of $\sum_{v \in [2]^{n_2}[3]^{n_3}}x(\{v\})$, where $x$ is a function $\mathcal{C}_k \rightarrow \mathbb{R}$ such that $x(\emptyset) = 1$ and $x(C) = 0$ if $C$ has minimum distance less than $d$, and such that the $S(D)\times S(D)$ matrix $(x(C\cup C'))_{C,C' \in S(D)}$ is positive semidefinite for each $D \in \mathcal{C}_k$. By exploiting symmetry, the semidefinite programming problem for the case $k=3$ is reduced using representation theory. It yields $135$ new upper bounds that are provided in tablesComment: 12 pages; some typos have been fixed. Accepted for publication in Discrete Mathematic

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers $q,n,d$, let $A_q(n,d)$ denote the maximum cardinality of a code of length $n$ over an alphabet $[q]$ with $q$ letters and with minimum distance at least $d$. We consider the following upper bound on $A_q(n,d)$. For any $k$, let \CC_k be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function \CC_4\to R_+ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, and such that the \CC_2\times\CC_2 matrix (x(C\cup C'))_{C,C'\in\CC_2} is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in $n$. It yields the new upper bounds $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$

Tutte's dichromate for signed graphs

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the ``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.Comment: 53 pp. 9 figure

Self-optimisation of admission control and handover parameters in LTE

In mobile cellular networks the handover (HO) algorithm is responsible for determining when calls of users that are moving from one cell to another are handed over from the former to the latter. The admission control (AC) algorithm, which is the algorithm that decides whether new (fresh or HO) calls that enter a cell are allowed to the cell or not, often tries to facilitate HO by prioritising HO calls in favour of fresh calls. In this way, a good quality of service (QoS) for calls that are already admitted to the network is pursued. In this paper, the effect of self-optimisation of AC parameters on the HO performance in a long term evolution (LTE) network is studied, both with and without the self-optimisation of HO parameters. Simulation results show that the AC parameter optimisation algorithm considerably improves the HO performance by reducing the amount of calls that are dropped prior to or during HO

The canonical Tutte polynomial for signed graphs

We construct a new polynomial invariant for signed graphs, the trivariate Tutte polynomial, which contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. While the Tutte polynomial of a graph is equivalently defined as the dichromatic polynomial or Whitney rank polynomial, the dichromatic polynomial of a signed graph (defined more widely for biased graphs by Zaslavsky) does not, by contrast, give the number of nowhere-zero flows as an evaluation in general. The trivariate Tutte polynomial contains Zaslavsky's dichromatic polynomial as a specialization. Furthermore, the trivariate Tutte polynomial gives as an evaluation the number of proper colorings of a signed graph under a more general sense of signed graph coloring in which colors are elements of an arbitrary finite set equipped with an involution.Peer ReviewedPostprint (published version

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let Aq(n, d) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on Aq(n, d). For any k, let Ck be the collection of codes of cardinality at most k. Then Aq(n, d) is at most the maximum value of Pv∈[q]n x({v}), where x is a function C4 → R+ such that x(∅) = 1 and x(C) = 0 if C has minimum distance less than d, and such that the C2 ×C2 matrix (x(C ∪C′))C,C′∈C2 is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds A4(6, 3) ≤ 176, A4(7, 4) ≤ 155, A5(7, 4) ≤ 489, and A5(7, 5) ≤ 87