arXiv subjects

Patrick Solé

Publications and source records attributed to Patrick Solé.

90 records · Page 5Linked to original sources

arXiv2012

The maximum number of minimal codewords in long codes

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide lower bounds. In this paper we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with $p$ vertices and $q$ edges can have only slightly more that $2^{q-p}$ cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than $2p+3\log_2(3p)$ edges. We also conclude that an Eulerian (even) graph has at most $2^{q-p}$ cycles unless the graph is a subdivision of a 4-regular graph that is the edge-disjoint union of two Hamiltonian cycles, in which case it may have as many as $2^{q-p}+p$ cycles.

cs.IT↗

arXiv2012

A new class of codes for Boolean masking of cryptographic computations

We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length $2n$ and dimension $n$ is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length $<132.$ We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths $\le 12$ by the building up construction. Some nonlinear permutations are constructed by using $\Z_4$-codes, based on the notion of dual distance of an unrestricted code.

cs.IT↗

arXiv2012

Cyclic codes over $M_2(\F_2)$

The ring in the title is the first non commutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of space time codes obtained by concatenation from the Golden code. In this article, we derive structure theorems for cyclic codes over that ring, and use them to characterize the lengths where self dual cyclic codes exist. These codes in turn give rise to formally self dual quaternary codes.

cs.IT↗

arXiv2011

Chebyshev's bias and generalized Riemann hypothesis

It is well known that $li(x)>π(x)$ (i) up to the (very large) Skewes' number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the specific distribution of non-trivial zeros $γ$ of the Riemann zeta function $ζ(s)$, encoded by the equation $li(x)-π(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_γ\frac{\sin (γ\log x)}γ]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $li[ψ(x)]>π(x)$ (ii) due to Robin \cite{Robin84}. A statement similar to (i) was found by Chebyshev that $π(x;4,3)-π(x;4,1)>0$ (iii) holds for any $x<26861$ \cite{Rubin94} (the notation $π(x;k,l)$ means the number of primes up to $x$ and congruent to $l\mod k$). The {\it Chebyshev's bias}(iii) is related to the generalized Riemann hypothesis (GRH) and occurs with a logarithmic density $\approx 0.9959$ \cite{Rubin94}. In this paper, we reformulate the Chebyshev's bias for a general modulus $q$ as the inequality $B(x;q,R)-B(x;q,N)>0$ (iv), where $B(x;k,l)=li[ϕ(k)*ψ(x;k,l)]-ϕ(k)*π(x;k,l)$ is a counting function introduced in Robin's paper \cite{Robin84} and $R$ resp. $N$) is a quadratic residue modulo $q$ (resp. a non-quadratic residue). We investigate numerically the case $q=4$ and a few prime moduli $p$. Then, we proove that (iv) is equivalent to GRH for the modulus $q$.

math.NT↗

arXiv2011

Classification of extremal and $s$-extremal binary self-dual codes of length 38

In this paper we classify all extremal and $s$-extremal binary self-dual codes of length 38. There are exactly 2744 extremal $[38,19,8]$ self-dual codes, two $s$-extremal $[38,19,6]$ codes, and 1730 $s$-extremal $[38,19,8]$ codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of $s$-extremal $[38,19,6]$ codes permits to achieve the classification of all $s$-extremal codes with d=6.

cs.DM↗

arXiv2011

Efficient prime counting and the Chebyshev primes

The function $ε(x)=\mbox{li}(x)-π(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $ε_θ(x)=\mbox{li}[θ(x)]-π(x)$ and $ε_ψ(x)=\mbox{li}[ψ(x)]-π(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $θ(x)=\sum_{p \le x} \log p$ and $ψ(x)=\sum_{n=1}^x Λ(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $μ(n)$ and $Λ(n)$ are the Möbius and the Von Mangoldt functions). Negative jumps in the above functions $ε$, $ε_θ$ and $ε_ψ$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{θ(p)}$ and $j_{ψ(p)}$. In particular, $j_p<1$, and $j_{θ(p)}>1$ for $p<10^{11}$. Besides, $j_{ψ(p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set $\mathcal{\mbox{Ch}}$, give accurate approximations of the jump $j_{ψ(p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $ψ(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $ψ(p_n^l)-p_n^l$ (or of the function $θ(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function $S_N(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{li}[ψ(x)^{1/n}]$, that is found to be much better than the standard Riemann prime counting function.

math.NT↗

arXiv2011

Constructive spherical codes near the Shannon bound

Shannon gave a lower bound in 1959 on the binary rate of spherical codes of given minimum Euclidean distance $ρ$. Using nonconstructive codes over a finite alphabet, we give a lower bound that is weaker but very close for small values of $ρ$. The construction is based on the Yaglom map combined with some finite sphere packings obtained from nonconstructive codes for the Euclidean metric. Concatenating geometric codes meeting the TVZ bound with a Lee metric BCH code over $GF(p),$ we obtain spherical codes that are polynomial time constructible. Their parameters outperform those obtained by Lachaud and Stern in 1994. At very high rate they are above 98 per cent of the Shannon bound.

cs.IT↗

arXiv2011

Pauli graphs, Riemann hypothesis, Goldbach pairs

Let consider the Pauli group $\mathcal{P}_q= $ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X|s> =|s+1>$ and $Z|s> =ω^s |s>$, with $ω=\exp(2iπ/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $ψ(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{ψ(q)}{q}>e^γ\log \log q$, involving the Euler constant $γ\sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q}$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^γ$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.

math-ph↗

arXiv2011

Riemann hypothesis and Quantum Mechanics

In their 1995 paper, Jean-Benoît Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function $ζ(β)$, where $β$ is an inverse temperature. We formulate Riemann hypothesis (RH) as a property of the low temperature Kubo-Martin-Schwinger (KMS) states of this theory. More precisely, the expectation value of the BC phase operator can be written as $$ϕ_β(q)=N_{q-1}^{β-1} ψ_{β-1}(N_q), $$ where $N_q=\prod_{k=1}^qp_k$ is the primorial number of order $q$ and $ ψ_b $ a generalized Dedekind $ψ$ function depending on one real parameter $b$ as $$ ψ_b (q)=q \prod_{p \in \mathcal{P,}p \vert q}\frac{1-1/p^b}{1-1/p}.$$ Fix a large inverse temperature $β>2.$ The Riemann hypothesis is then shown to be equivalent to the inequality $$ N_q |ϕ_β(N_q)|ζ(β-1) >e^γ\log \log N_q, $$ for $q$ large enough. Under RH, extra formulas for high temperatures KMS states ($1.5< β<2$) are derived.

math-ph↗

arXiv2011

Extreme values of the Dedekind $Ψ$ function

Let $Ψ(n):=n\prod_{p | n}(1+\frac{1}{p})$ denote the Dedekind $Ψ$ function. Define, for $n\ge 3,$ the ratio $R(n):=\frac{Ψ(n)}{n\log\log n}.$ We prove unconditionally that $R(n)< e^γ$ for $n\ge 31.$ Let $N_n=2...p_n$ be the primorial of order $n.$ We prove that the statement $R(N_n)>\frac{e^γ}{ζ(2)}$ for $n\ge 3$ is equivalent to the Riemann Hypothesis.

math.NT↗

arXiv2010

On Nicolas criterion for the Riemann Hypothesis

Nicolas criterion for the Riemann Hypothesis is based on an inequality that Euler totient function must satisfy at primorial numbers. A natural approach to derive this inequality would be to prove that a specific sequence related to that bound is strictly decreasing. We show that, unfortunately, this latter fact would contradict Cramér conjecture on gaps between consecutive primes. An analogous situation holds when replacing Euler totient by Dedekind $Ψ$ function.

math.NT↗

arXiv2010

Robin inequality for $7-$free integers

Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some prime $p.$ We give a method to check Robin inequality $σ(n) < e^γn\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce $Ψ_t,$ a generalization of Dedekind $Ψ$ function defined for any integer $t\ge 2$ by $$Ψ_t(n):=n\prod_{p | n}(1+1/p+...+1/p^{t-1}).$$ If $n$ is $t-$free then the sum of divisor function $σ(n)$ is $ \le Ψ_t(n).$ We characterize the champions for $x \mapsto Ψ_t(x)/x,$ as primorial numbers. Define the ratio $R_t(n):=\frac{Ψ_t(n)}{n\log\log n}.$ We prove that, for all $t$, there exists an integer $n_1(t),$ such that we have $R_t(N_n)< e^γ$ for $n\ge n_1,$ where $N_n=\prod_{k=1}^np_k.$ Further, by combinatorial arguments, this can be extended to $R_t(N)\le e^γ$ for all $N\ge N_n,$ such that $n\ge n_1(t).$ This yields Robin inequality for $t=6,\,7.$ For $t$ varying slowly with $N$, we also derive $R_t(N)< e^γ.$

math.NT↗

arXiv2010

A Type II lattice of norm 8 in dimension 72

A Type II lattice of norm 8 in dimension 72 is obtained by Construction A applied to an extended Quadratic Residue code over Z8. Its automorphism group contains a subgroup isomorphic to PSL(2,71).

cs.IT↗

arXiv2010

An extension of Massey scheme for secret sharing

We consider an extension of Massey's construction of secret sharing schemes using linear codes. We describe the access structure of the scheme and show its connection to the dual code. We use the $g$-fold weight enumerator and invariant theory to study the access structure.

cs.IT↗

arXiv2008

Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.

quant-ph↗

arXiv2008

Pseudo-random Sequences Generated by Cellular Automata

Generation of pseudo random sequences by cellular automata, as well as by hybrid cellular automata is surveyed. An application to the fast evaluation and FPGA implementation of some classes of boolean functions is sketched out.

cs.DM↗

arXiv2002

The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is self-dual -- which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z_4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z_4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z_4-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.

math.CO↗

arXiv1993

A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over $\ZZ_4$, the integers $\bmod~4$. The Kerdock and Preparata codes are duals over $\ZZ_4$, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over $\ZZ_4$. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over $\ZZ_4$, but Hamming codes in general are not, nor is the Golay code.

math.CO↗