Linear Forms and Higher-Degree Uniformity for Functions On {\mathbb{F}^{n}_{p}}

In [GW1] we began an investigation of the following general question. Let L ₁, . . . , L _m be a system of linear forms in d variables on Fⁿ_p{F^n_p}, and let A be a subset of Fⁿ_p{F^n_p} of positive density. Under what circumstances can one prove that A contains roughly the same number of m-tuples L ₁(x ₁, . . . , x _d ), . . . , L _m (x ₁, . . . , x _d ) with x₁,?, x_d ? \mathbb Fⁿ_p{x_1,\ldots, x_d \in {\mathbb F}^n_p} as a typical random set of the same density? Experience with arithmetic progressions suggests that an appropriate assumption is that ||A - d1||_U^k{||A - \delta 1||_{U{^k}}} should be small, where we have written A for the characteristic function of the set A, δ is the density of A, k is some parameter that depends on the linear forms L ₁, . . . , L _m , and || ·||_U^k{|| \cdot ||_U{^k}} is the kth uniformity norm. The question we investigated was how k depends on L ₁, . . . , L _m . Our main result was that there were systems of forms where k could be taken to be 2 even though there was no simple proof of this fact using the Cauchy–Schwarz inequality. Based on this result and its proof, we conjectured that uniformity of degree k − 1 is a sufficient condition if and only if the kth powers of the linear forms are linearly independent. In this paper we prove this conjecture, provided only that p is sufficiently large. (It is easy to see that some such restriction is needed.) This result represents one of the first applications of the recent inverse theorem for the U ^k norm over Fⁿ_p{F^n_p} by Bergelson, Tao and Ziegler [TZ2], [BTZ]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.     

Linear codes over {\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}

In this work, we investigate linear codes over the ring ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linearity of binary codes under the Gray map and give a main class of binary codes as an example of ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes. The duals and the complete weight enumerators for ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ are obtained.     

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

Navigating in the Cayley Graphs of {\rm SL}_{N}(\mathbb{Z}) and {\rm SL}_{N}(\mathbb{F}_{p})

We give a nondeterministic algorithm that expresses elements of , for N ≥ 3, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the nondeterministic time-complexity of the subtractive version of Euclid’s algorithm for finding the greatest common divisor of N ≥ 3 integers a₁, ..., a_N is at most a constant times . This leads to an elementary proof that for N ≥ 3 the word metric in is biLipschitz equivalent to the logarithm of the matrix norm – an instance of a theorem of Mozes, Lubotzky and Raghunathan. And we show constructively that there exists K>0 such that for all N ≥ 3 and primes p, the diameter of the Cayley graph of with respect to the generating set is at most .Mathematics Subject Classification: 20F05     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      6. Burgess Inequality In {\mathbb {F}_{p^2}}      Mei-Chu Chang 《Geometric And Functional Analysis》2009,19(4):1001-1016 The purpose of the paper is to present new estimates on incomplete character sums in finite fields that are of the strength of Burgess’ celebrated theorem for prime fields. More precisely, an inequality of this type is obtained in Fp2{F_{p^2}} and also for binary quadratic forms, improving on the work of Davenport–Lewis and on several results due to Burgess. The arguments are based on new estimates for the multiplicative energy of certain sets that allow us to improve the amplification step in Burgess’ method.      7. Additive semisimple multivariable codes over {\mathbb{F}_4}      E. Martínez-Moro  A. Piñera-Nicolás  I. F. Rúa 《Designs, Codes and Cryptography》2013,69(2):161-180 The structure of additive multivariable codes over ${\mathbb{F}_4}$ (the Galois field with 4 elements) is presented. The semisimple case (i.e., when the defining polynomials of the code have no repeated roots) is specifically addressed. These codes extend in a natural way the abelian codes, of which additive cyclic codes of odd length are a particular case. Duality of these codes is also studied.      8. Maximum distance separable codes over {\mathbb{Z}_4} and {\mathbb{Z}_2 \times \mathbb{Z}_4}      M. Bilal  J. Borges  S. T. Dougherty  C. Fern��ndez-C��rdoba 《Designs, Codes and Cryptography》2011,61(1):31-40 Known upper bounds on the minimum distance of codes over rings are applied to the case of ${\mathbb Z_{2}\mathbb Z_{4}}$ -additive codes, that is subgroups of ${\mathbb Z_{2}^{\alpha}\mathbb Z_{4}^{\beta}}$ . Two kinds of maximum distance separable codes are studied. We determine all possible parameters of these codes and characterize the codes in certain cases. The main results are also valid when ?? = 0, namely for quaternary linear codes.      9. Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}      Jacques Wolfmann 《Designs, Codes and Cryptography》2000,20(1):73-88 Let R=GR(4,m) be the Galois ring of cardinality 4m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ λ Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ λ Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D i as the set of x in R such thatφ λ Π =i. We prove the following results: 1) D i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D i and the set of φ(D i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D i in the additive group of R. Examples are given by means of morphisms and norm in R.      10. Towards factoring in {SL(2,\,\mathbb{F}_{2^n})}      Christophe Petit 《Designs, Codes and Cryptography》2014,71(3):409-431 The security of many cryptographic protocols relies on the hardness of some computational problems. Besides discrete logarithm or integer factorization, other problems are regularly proposed as potential hard problems. The factorization problem in finite groups is one of them. Given a finite group G, a set of generators generators for this group and an element ${g\in G}$ , the factorization problem asks for a “short” representation of g as a product of the generators. The problem is related to a famous conjecture of Babai on the diameter of Cayley graphs. It is also motivated by the preimage security of Cayley hash functions, a particular kind of cryptographic hash functions. The problem has been solved for a few particular generator sets, but essentially nothing is known for generic generator sets. In this paper, we make significant steps towards a solution of the factorization problem in the group ${G:=\,SL(2,\,\mathbb{F}_{2^n})}$ , a particularly interesting group for cryptographic applications. To avoid considering all generator sets separately, we first give a new reduction tool that allows focusing on some generator sets with a “nice” special structure. We then identify classes of trapdoor matrices for these special generator sets, such that the factorization of a single one of these matrices would allow efficiently factoring any element in the group. Finally, we provide a heuristic subexponential time algorithm that can compute subexponential length factorizations of any element for any pair of generators of ${SL(2,\,\mathbb{F}_{2^n})}$ . Our results do not yet completely remove the factorization problem in ${SL(2,\,\mathbb{F}_{2^n})}$ from the list of potential hard problems useful for cryptography. However, we believe that each one of our individual results is a significant step towards a polynomial time algorithm for factoring in ${SL(2,\,\mathbb{F}_{2^n})}$ .      11. Semisimple multivariable mathbb {F}_q-linear codes over mathbb {F}_{q^l}      Yonglin Cao  Jian Gao  Fang-Wei Fu 《Designs, Codes and Cryptography》2015,77(1):153-177       12. Torified varieties and their geometries over {\mathbb{F}_1}      Javier L��pez Pe?a  Oliver Lorscheid 《Mathematische Zeitschrift》2011,267(3-4):605-643 This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over ${\mathbb Z}$ that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes?CConsani and an object in the sense of Soulé and show that both are varieties over ${\mathbb{F}_1}$ in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over ${\mathbb{F}_1}$ in the literature so far. Furthermore, we compare Connes?CConsani??s geometry, Soulé??s geometry and Deitmar??s geometry, and we discuss to what extent Chevalley groups can be realized as group objects over ${\mathbb{F}_1}$ in the given categories.      13. Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$      Hai Q. Dinh  Abhay Kumar Singh  Sukhamoy Pattanayak  Songsak Sriboonchitta 《Designs, Codes and Cryptography》2018,86(7):1451-1467 We study the structure of cyclic DNA codes of odd length over the finite commutative ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2 + v^2\mathbb {F}_2+uv^2\mathbb {F}_2,~u^2=0, v^3=v\), which plays an important role in genetics, bioengineering and DNA computing. A direct link between the elements of the ring R and 64 codons used in the amino acids of living organisms is established by introducing a Gray map from R to \(R_1=\mathbb {F}_2+u\mathbb {F}_2 ~(u^2=0)\). The reversible and the reversible-complement codes over R are investigated. We also discuss the binary image of the cyclic DNA codes over R. Among others, some examples of DNA codes obtained via Gray map are provided.      14. New Linear Codes Over \mathbb{F}_3 and \mathbb{F}_5 and Improvements on Bounds      Irfan Siap  Dijen K. Ray-Chaudhuri 《Designs, Codes and Cryptography》2000,21(1-3):223-233 One of the most important problems of coding theory is to constructcodes with best possible minimum distances. In this paper, we generalize the method introduced by [8] and obtain new codes which improve the best known minimum distance bounds of some linear codes. We have found a new linear ternary code and 8 new linear codes over with improved minimumdistances. First we introduce a generalized version of Gray map,then we give definition of quasi cyclic codes and introduce nearlyquasi cyclic codes. Next, we give the parameters of new codeswith their generator matrices. Finally, we have included twotables which give Hamming weight enumerators of these new codes.      15. On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$      A. Melakhessou  K. Guenda  T. A. Gulliver  M. Shi  P. Solé 《Journal of Applied Mathematics and Computing》2018,57(1-2):375-391 In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.      16. Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes      Marcelo Muniz 《Designs, Codes and Cryptography》2006,41(2):147-152 Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).      17. On some permutation binomials and trinomials over $$\mathbb {F}_{2^n}$$      Srimanta Bhattacharya  Sumanta Sarkar 《Designs, Codes and Cryptography》2017,82(1-2):149-160 In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.      18. On the Number of Trace-One Elements in Polynomial Bases for \mathbb{F}_{2^n}      Omran Ahmadi  Alfred Menezes 《Designs, Codes and Cryptography》2005,37(3):493-507 This paper investigates the number of trace-one elements in a polynomial basis for . A polynomial basis with a small number of trace-one elements is desirable because it results in an efficient and low cost implementation of the trace function. We focus on the case where the reduction polynomial is a trinomial or a pentanomial, in which case field multiplication can also be efficiently implemented. Communicated by: P. Wild      19. The Second Variational Formula for Laguerre Minimal Hypersurfaces in {\mathbb {R}^n}      Yuping Song 《Results in Mathematics》2013,63(3-4):985-998 Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in ${\mathbb{R}^{n}}$ and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in ${\mathbb{R}^{n}}$ and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in ${\mathbb{R}^{3}_{0}}$ is Laguerre equivalent to a stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ under the Laguerre embedding. This example of stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).      20. $${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel      Cristina Fernández-Córdoba  Jaume Pujol  Mercè Villanueva 《Designs, Codes and Cryptography》2010,56(1):43-59 A code C{{\mathcal C}} is \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.

Burgess Inequality In {\mathbb {F}_{p^2}}

The purpose of the paper is to present new estimates on incomplete character sums in finite fields that are of the strength of Burgess’ celebrated theorem for prime fields. More precisely, an inequality of this type is obtained in F_p²{F_{p^2}} and also for binary quadratic forms, improving on the work of Davenport–Lewis and on several results due to Burgess. The arguments are based on new estimates for the multiplicative energy of certain sets that allow us to improve the amplification step in Burgess’ method.     

Additive semisimple multivariable codes over {\mathbb{F}_4}

The structure of additive multivariable codes over ${\mathbb{F}_4}$ (the Galois field with 4 elements) is presented. The semisimple case (i.e., when the defining polynomials of the code have no repeated roots) is specifically addressed. These codes extend in a natural way the abelian codes, of which additive cyclic codes of odd length are a particular case. Duality of these codes is also studied.     

Maximum distance separable codes over {\mathbb{Z}_4} and {\mathbb{Z}_2 \times \mathbb{Z}_4}

Known upper bounds on the minimum distance of codes over rings are applied to the case of ${\mathbb Z_{2}\mathbb Z_{4}}$ -additive codes, that is subgroups of ${\mathbb Z_{2}^{\alpha}\mathbb Z_{4}^{\beta}}$ . Two kinds of maximum distance separable codes are studied. We determine all possible parameters of these codes and characterize the codes in certain cases. The main results are also valid when ?? = 0, namely for quaternary linear codes.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Towards factoring in {SL(2,\,\mathbb{F}_{2^n})}

The security of many cryptographic protocols relies on the hardness of some computational problems. Besides discrete logarithm or integer factorization, other problems are regularly proposed as potential hard problems. The factorization problem in finite groups is one of them. Given a finite group G, a set of generators generators for this group and an element ${g\in G}$ , the factorization problem asks for a “short” representation of g as a product of the generators. The problem is related to a famous conjecture of Babai on the diameter of Cayley graphs. It is also motivated by the preimage security of Cayley hash functions, a particular kind of cryptographic hash functions. The problem has been solved for a few particular generator sets, but essentially nothing is known for generic generator sets. In this paper, we make significant steps towards a solution of the factorization problem in the group ${G:=\,SL(2,\,\mathbb{F}_{2^n})}$ , a particularly interesting group for cryptographic applications. To avoid considering all generator sets separately, we first give a new reduction tool that allows focusing on some generator sets with a “nice” special structure. We then identify classes of trapdoor matrices for these special generator sets, such that the factorization of a single one of these matrices would allow efficiently factoring any element in the group. Finally, we provide a heuristic subexponential time algorithm that can compute subexponential length factorizations of any element for any pair of generators of ${SL(2,\,\mathbb{F}_{2^n})}$ . Our results do not yet completely remove the factorization problem in ${SL(2,\,\mathbb{F}_{2^n})}$ from the list of potential hard problems useful for cryptography. However, we believe that each one of our individual results is a significant step towards a polynomial time algorithm for factoring in ${SL(2,\,\mathbb{F}_{2^n})}$ .     

Torified varieties and their geometries over {\mathbb{F}_1}

This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over ${\mathbb Z}$ that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes?CConsani and an object in the sense of Soulé and show that both are varieties over ${\mathbb{F}_1}$ in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over ${\mathbb{F}_1}$ in the literature so far. Furthermore, we compare Connes?CConsani??s geometry, Soulé??s geometry and Deitmar??s geometry, and we discuss to what extent Chevalley groups can be realized as group objects over ${\mathbb{F}_1}$ in the given categories.     

Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$

We study the structure of cyclic DNA codes of odd length over the finite commutative ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2 + v^2\mathbb {F}_2+uv^2\mathbb {F}_2,~u^2=0, v^3=v\), which plays an important role in genetics, bioengineering and DNA computing. A direct link between the elements of the ring R and 64 codons used in the amino acids of living organisms is established by introducing a Gray map from R to \(R_1=\mathbb {F}_2+u\mathbb {F}_2 ~(u^2=0)\). The reversible and the reversible-complement codes over R are investigated. We also discuss the binary image of the cyclic DNA codes over R. Among others, some examples of DNA codes obtained via Gray map are provided.     

New Linear Codes Over \mathbb{F}_3 and \mathbb{F}_5 and Improvements on Bounds

One of the most important problems of coding theory is to constructcodes with best possible minimum distances. In this paper, we generalize the method introduced by [8] and obtain new codes which improve the best known minimum distance bounds of some linear codes. We have found a new linear ternary code and 8 new linear codes over with improved minimumdistances. First we introduce a generalized version of Gray map,then we give definition of quasi cyclic codes and introduce nearlyquasi cyclic codes. Next, we give the parameters of new codeswith their generator matrices. Finally, we have included twotables which give Hamming weight enumerators of these new codes.     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes

Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).     

On some permutation binomials and trinomials over $$\mathbb {F}_{2^n}$$

In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.     

On the Number of Trace-One Elements in Polynomial Bases for \mathbb{F}_{2^n}

This paper investigates the number of trace-one elements in a polynomial basis for . A polynomial basis with a small number of trace-one elements is desirable because it results in an efficient and low cost implementation of the trace function. We focus on the case where the reduction polynomial is a trinomial or a pentanomial, in which case field multiplication can also be efficiently implemented. Communicated by: P. Wild     

The Second Variational Formula for Laguerre Minimal Hypersurfaces in {\mathbb {R}^n}

Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in ${\mathbb{R}^{n}}$ and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in ${\mathbb{R}^{n}}$ and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in ${\mathbb{R}^{3}_{0}}$ is Laguerre equivalent to a stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ under the Laguerre embedding. This example of stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.