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We consider the Gaussian wiretap channel, where two legitimate players Alice and Bob communicate over an additive white Gaussian noise (AWGN) channel, while Eve is eavesdropping, also through an AWGN channel. We propose a coding strategy based on lattice coset encoding. We define the secrecy gain as a design criterion for wiretap lattice codes, expressed in terms of the lattice theta series, which characterizes Eve’s confusion as a function of the channel parameters. The secrecy gain is studied for even unimodular lattices, and an asymptotic analysis shows that it grows exponentially in the dimension of the lattice. Examples of wiretap lattice codes are given. [ABSTRACT FROM AUTHOR]

In this paper, we present an analog of the Singleton bound for convolutional codes over finite rings. Codes meeting that bound are called MDS. A constructive method for the MDS codes, restricted to free codes, is presented via cyclic codes over finite rings. Examples of nonfree MDS codes are provided. [ABSTRACT FROM AUTHOR]

It is known that, for transmission over quasi-static MIMO fading channels with n transmit antennas, diversity can be obtained by using an inner fully diverse space-time block code while coding gain, derived from the determinant criterion, comes from an appropriate outer code. When the inner code has a cyclic algebra structure over a number field, as for perfect space-time codes, an outer code can be designed via coset coding, more precisely, by taking the quotient of the algebra by a two-sided ideal which leads to matrices over finite alphabets for the outer code. In this paper, we show that the determinant criterion induces various metrics on the outer code, such as the Hamming and Bachoc distances. When n=2, partitioning the 2\,\times\,2 Golden code by using an ideal above the prime 2 leads to consider codes over either \cal M2(\BBF2) or \cal M2(\BBF2[i]), both being noncommutative alphabets. By identifying them as algebras over a finite field or a finite ring respectively, we establish an unexpected connection with classical error-correcting codes over \BBF4 and \BBF4[i]. Matrix rings of higher dimension, suitable for 3\,\times\,3 and 4\,\times\,4 perfect codes, give rise to more complex examples. [ABSTRACT FROM AUTHOR]

Families of binary low-correlation sequences with high nonlinearity (in relation with their Walsh Hadamard transform) are constructed by using the most significant bit of linear recurrence sequences over the ring ℤ2l, for l ≥ 3. The engineering motivation is the design of a multiple-code code-division multiple-access (CDMA) scheme with a control of low peak-to-average power ratio (PAPR). Proof techniques combine Galois ring theory (local Weil bound) with spectral analysis over the additive group of ℤ2l. New estimates on the size of weighted degree trace codes are derived. The parameters of the sequences families constructed are shown to lie above a modified Varshamov-Gilbert bound. [ABSTRACT FROM AUTHOR]

Presents information on a study which described an algebraic approach to quasi-cyclic codes. Some basic notations and definitions; Trace formula for quasi-cyclic codes; Conclusions.

Focuses on a study which investigated duadic codes which constitute a family of binary cyclic codes. Interest in codes over finite rings; Theorems and proofs on duadic codes; Augmented and extended Abelian codes.

Presents a study which focused on the towers of function fields meeting with Drinfeld-Vladut bound with equality. Information on the hypergeometric analogs of the Arithmetic-Geometric Mean; Interpretation of the algorithms used.

Introduces the type II Z4-codes as self-dual codes over the integers modulo 4 containing the all-one vector with Euclidean weights multiple of eight. Invariant theory of the self-dual codes; Type II Z4-codes; Examples of families of type II codes with their general construction; Even unimodular lattices.

The imbalance and the autocorrelation of the binary sequences in the title are explored by combining the local Weil bound with spectral analysis. Recent estimates on these quantities are improved by a factor of order 2½, for large l. [ABSTRACT FROM AUTHOR]

We investigate fat projective linear codes over ${\mathbb Z}_{p^{m}}$ , $m\geqslant 2$ , with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator matrix of the code determine a cap in the projective Hjelmslev geometry $\mathop {\mathrm {PHG}}\nolimits (k-1, {\mathbb Z}_{p^{m}})$. This generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982). The proof relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry. [ABSTRACT FROM AUTHOR]

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the $\mathbb {Z}_{2} \mathbb {Z}_{4}$ -additive quasi-cyclic codes. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function $\Gamma _{Q}(n,\ell,k,q)$ , that is defined to be the largest number of nonzero weights a quasi-cyclic code of index $\gcd (\ell,n)$ , length $n$ and dimension $k$ over $\mathbb F_{q}$ can have, and connect it to similar functions related to linear and cyclic codes. We provide several upper and lower bounds on this function, using different techniques and studying its asymptotic behavior. Moreover, we determine the smallest index for which a $q$ -ary Reed-Muller code is quasi-cyclic, a result of independent interest. [ABSTRACT FROM AUTHOR]

The dual of the Kasami code of length $q^{2}-1$ , with $q$ a power of 2, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_{q}$ with a Simplex code of length $q-1$. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related $q$ -ary MDS code are determined. New cyclic completely regular codes over finite fields a power of 2, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code. [ABSTRACT FROM AUTHOR]

The construction of deletion codes for the editing metric is reduced to the construction of codes over the integers for the Manhattan metric by run length coding. The latter codes are constructed by expurgation of lattices’ translates. These lattices, in turn, are obtained from Construction A applied to binary codes and \mathbb Z4 –codes. A lower bound on the size of our codes for the Manhattan distance are obtained through generalized theta series of the corresponding lattices. For any fixed number of deletions, provided the number of runs is large enough our method supplies a correction technique. For fixed number of runs and binary sequence length large our lattice construction is shown to be tight up to constants. [ABSTRACT FROM PUBLISHER]

We construct two new infinite families of trace codes of dimension 2m , over the ring \mathbb {F}_{p}+u\mathbb {F}_{p} , with u^{2}=u , when p is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear p -ary codes of respective lengths and 2(p^m-1)^2 . When m is singly even, the first family gives five-weight codes. When m is odd and p\equiv 3 \pmod {4} , the first family yields p$ -ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever $p=3$ and $m \ge 3$ , or $p\ge 5$ and $m\ge 4$ . Applications to secret sharing schemes are given. [ABSTRACT FROM PUBLISHER]

We prove that the covering radius of the Melas code $M(m,q)$ of length $n=q^{m}-1$ over $\mathbb {F}_{q}$ is 2 if $q > 3$. We also prove that the covering radius of $M(m,3)$ is 3 is $m \ge 3$ , the covering radius of $M(2,3)$ is 4, and the covering radii of $M(1,2)$ and $M(1,3)$ are 1. [ABSTRACT FROM AUTHOR]

Hermitian self-dual abelian codes in a group ring \BBFq^{2}[G], where \BBFq^{2} is a finite field of order q^2 and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semisimple group ring, a characterization of Hermitian self-dual abelian codes in \BBFq^{2}[G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in \BBFq^{2}[G], i.e., there exists a Hermitian self-dual abelian code in \BBFq^{2}[G] if and only if the order of G is even and q=2^l for some positive integer l. Later on, the study is further restricted to the case where \BBF2^{2l}[G] is a principal ideal group ring, or equivalently, G\cong A\oplus\BBZ2^{k} with 2\nmid\vert A\vert. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in \BBF{2^{2l}}[A\oplus\BBZ2^{k}] can be determined easily. When A is cyclic, this result answers an open problem of Jia concerning Hermitian self-dual cyclic codes. In many cases, \BBF{2^{2l}}[A\oplus\BBZ2^{k}] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in \BBF{2^{2l}}[A\oplus\BBZ2] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2 m over \BBF{2^{2l}} exists. [ABSTRACT FROM AUTHOR]

We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermitian inner product. Various families of additive codes over \BBF 4 are used in the construction of many asymmetric quantum codes over \BBF 4. [ABSTRACT FROM PUBLISHER]

The weights in maximum distance separable (MDS) codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n-k+1 to n. The proof uses the covering radius of the dual code. [ABSTRACT FROM AUTHOR]

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function $\Gamma (k,q)$ , that is defined as the largest number of nonzero weights a cyclic code of dimension $k$ over $\mathbb {F}_{q}$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or $q$ -ary, as well as in the $q$ -ary Hamming code are determined, two difficult results of independent interest. [ABSTRACT FROM AUTHOR]

The 1+/n+ method, based on an ECRIS charge breeder (CB) originally developed at the LPSC laboratory, is now implemented at GANIL for the production of Radioactive Ion Beams (RIBs). Prior to its installation in the middle of the low energy beam line of the SPIRAL1 facility, the 1+/n+ system CB has been modified based on the experiments performed on the CARIBU Facility at Argone National Laboratory. Later, it has been tested at the 1+/n+ LPSC test bench to validate its operation performances. Charge breeding efficiencies as well as charge breeding times have been measured for noble gases and alkali elements. The commissioning phase started at GANIL in the second half-year of 2017. It consisted of a stepwise process to test the upgrade of the SPIRAL1 facility from simple validation [operation of Charge Breeder (CB) as a stand-alone source] up to the production of the first 1+/n+ RIB. Thus, this year, a 38mK/38K RIB has been successfully delivered to a physics experiment over a period of 1 week. The yields on the physics target were in the range of ∼2–4 × 106 pps at 9 MeV/u. The target ion source system (TISS) was made of a FEBIAD ion source connected to a hot graphite target. This is the first time a RIB is accelerated with a cyclotron with the 1+/n+ method. Moreover, a production test with the FEBIAD TISS has confirmed the yields measured previously, which validates the extension of the GANIL/SPIRAL1 catalog for a number of isotopes. In parallel, R&D is being performed on new TISSs (e.g., a fast release one, using surface ionization source). Targets are also a subject of ongoing R&D for yield and release time optimization. This contribution will present the new acceleration scheme of the SPIRAL1 facility, which largely extends the palette of RIBs available for nuclear physicists. It will be compared to the ones used at similar ISOL facilities. This facility is more than a simple ISOL facility, and an overview of the new opportunities offered by the upgraded installation will be also discussed. [ABSTRACT FROM AUTHOR]

We study linear complementary pairs (LCP) of codes $(C, D)$ , where both codes belong to the same algebraic code family. We especially investigate constacyclic and quasi-cyclic LCP of codes. We obtain characterizations for LCP of constacyclic codes and LCP of quasi-cyclic codes. Our result for the constacyclic complementary pairs extends the characterization of linear complementary dual (LCD) cyclic codes given by Yang and Massey. We observe that when $C$ and $D$ are complementary and constacyclic, the codes $C$ and $D^\bot $ are equivalent to each other. Hence, the security parameter $\min (d(C),d(D^\bot))$ for LCP of codes is simply determined by one of the codes in this case. The same holds for a special class of quasi-cyclic codes, namely 2D cyclic codes, but not in general for all quasi-cyclic codes, since we have examples of LCP of double circulant codes not satisfying this conclusion for the security parameter. We present examples of binary LCP of quasi-cyclic codes and obtain several codes with better parameters than known binary LCD codes. Finally, a linear programming bound is obtained for binary LCP of codes and a table of values from this bound is presented in the case $d(C)=d(D^\bot)$. This extends the linear programming bound for LCD codes. [ABSTRACT FROM AUTHOR]

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function response, and extend classical coding methods to construct multiply constant-weight codes from known \(q\) -ary and constant-weight codes. We derive analogs of Johnson bounds and give constructions showing these bounds to be asymptotically tight up to a constant factor under certain conditions. We also examine the rates of multiply constant-weight codes and demonstrate that these rates are the same as those of constant-weight codes of corresponding parameters. [ABSTRACT FROM PUBLISHER]

We introduce complementary information set codes of higher order. A binary linear code of length \(tk\) and dimension \(k\) is called a complementary information set code of order \(t\) ( \(t\) -CIS code for short) if it has \(t\) pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a \(t\) -CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order 3 is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length \(\le 12\) is given. Nonlinear codes better than linear codes are derived by taking binary images of \( {\mathbb Z}_{4}\) -codes. A general algorithm based on Edmonds’ basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate \(1/t\) , it either provides \(t\) disjoint information sets or proves that the code is not \(t\) -CIS. Using this algorithm, all optimal or best known \([tk, k]\) codes, where \(t=3, 4, {\dots }, 256\) and \(1 \le k \le \lfloor 256/t \rfloor \) are shown to be \(t\) -CIS for all such \(k\) and \(t\) , except for \(t=3\) with \(k=44\) and \(t=4\) with \(k=37\) . [ABSTRACT FROM AUTHOR]

We introduce a new class of rate one-half binary codes: 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 vectorial 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 \leq 12 by the building up construction. Some nonlinear permutations are constructed by using \BBZ4-codes, based on the notion of dual distance of a possibly nonlinear code. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The maximum size of a binary code is studied as a function of its length n, minimum distance d, and minimum codeword weight \ssi w. This function B(n,d,\ssi w) is first characterized in terms of its exponential growth rate in the limit n\rightarrow\infty for fixed \delta=d/n and \omega=\ssi w/n. The exponential growth rate of B(n,d,\ssi w) is shown to be equal to the exponential growth rate of A(n,d) for 0\leq\omega\leq 1/2, and equal to the exponential growth rate of A(n,d,\ssi w) for 1/2<\omega\leq 1. Second, analytic and numerical upper bounds on B(n,d,\ssi w) are derived using the semidefinite programming (SDP) method. These bounds yield a nonasymptotic improvement of the second Johnson bound and are tight for certain values of the parameters. [ABSTRACT FROM AUTHOR]

We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 3l as codes of length l over the ring F[sub 2] × F[sub 4]. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New extremal binary codes in lengths ≤ 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72, 36, 16] Type II code are derived. [ABSTRACT FROM AUTHOR]

We will present recent basic developments about possible extension of the COMIC (for COmpact MIcrowave and Coaxial) devices up to 5.8 GHz in place of the present 2.45 GHz operation [P. Sortais, T. Lamy, J. Médard, J. Angot, L. Latrasse, and T. Thuillier, Rev. Sci. Instrum. 81, 02B314 (2010)]. New applications associating multiple COMIC devices for thin film deposition will be described and we will explain why an increase of the current density delivered by each individual ion source could lead to the increase of the deposition rate. For this purpose, we will present results of about two devices working at 5.8 GHz. The first one is a tiny ion source, the world smallest microwave ion source, exactly similar to COMIC but operating at 5.8 GHz with a quarter wave cavity structure and a few watts microwave power consumption. We will show that the frequency scaling effect is effective inside such small machines. The second one is a more ambitious ion source designed around a three quarter wave structure that works with a few tens of watts at 5.8 GHz. [ABSTRACT FROM AUTHOR]