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... Functions between two Abelian finite groups A and B with the same cardinality and the smallest possible nonlinearity are called perfect nonlinear (PN) functions. These functions are related to Sidon sets (a subset of an additive group with the property that all sums of two elements in this set are different), since a function is APN if and only if its associated graph is a Sidon set in the product group A × B. This is one of the reasons we focus on an algebraic construction to introduce new cryptographic functions that arise from the construction of a finite Sidon set due to Bose [8]. The main contribution of this paper is the construction of a new set of functions with good linear and differential properties based on the construction of Sidon sets. ...
... In this section, we present the construction of our function, which is based on the construction of Sidon sets due to Bose [8]. ...
... Sidon sets can also be defined as those with the property that all nonzero differences of pairs of elements are distinct. In particular, the construction of Sidon sets due to Bose is as follows [8]. ...
Sidon sets have several applications in mathematics and in real-world problems, including the generation of secret keys in cryptography, error-correcting codes, and the physical problem of compression of signals in telecommunications. In particular, in cryptography, the design of cryptographic functions with optimal properties like nonlinearity and differential uniformity plays a fundamental role in the development of secure cryptographic systems. Based on the construction of Bose-type Sidon sets, in this paper we present the construction of a new cryptographic function with good properties of nonlinearity and differential uniformity.
... We next develop constructions in general finite groups based on relative difference sets (RDSs), in which the subgroup not present in the union of the sets of the DPDF/EPDF is precisely the forbidden subgroup for the RDS. In particular, we show how the classic result of Bose [1] which originally constructed relative difference sets using finite geometry, very naturally extends to a DPDF/EPDF construction in cyclic groups. We obtain a framework for using RDSs for DPDF/EPDF constructions which encompasses this example and generates many others. ...
... Relative difference sets were first introduced by Bose in [1], though they were not named as such; he presented his result as the "affine analogue" of Singer's Theorem on difference sets. The name and concept of RDS were formally introduced by Elliott and Butson in [8]. ...
... A parallel class has q lines, each with q points, and the lines in the class partition the points of the affine plane. From Bose's paper, [1], each line with c = 0 in the parallel class gives an RDS with the required parameters, and taking all q − 1 such lines, we obtain the DPDF described. To see this directly, replace each point in AG(2, q) by the corresponding power of α via the above identification. ...
Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields. We present the first non-cyclotomic infinite families of DPDFs which are also EPDFs, in structures other than finite fields (in particular cyclic groups and non-abelian groups). As well as direct constructions, we present an approach to constructing DPDFs/EPDFs using relative difference sets (RDSs); as part of this, we demonstrate how the well-known RDS result of Bose extends to a very natural construction for DPDFs and EPDFs.
... If we remove any conditioning on the field size, then there do exist codes that are (n−k, L)-list decodable for some L < n k . For example, the linear [8,4] code over GF (7) in Example 2 below is (4, 50)-list decodable, and the linear [6,2] code over GF (5) in Example 6 below is (4, 11)list decodable. ...
... and every L ∈ Z + . 6 The same holds for a similar construction of length n = 7 over F = GF(73), where (α j ) j∈[n] = (0 1 9 8 3 16 34). ...
... The following lemma provides a necessary and sufficient condition for a linear MDS code C to be lightly-L-MDS. The condition is expressed in terms of the non-singularity of matrices M = M J0,J1,...,JL (H) (as in (7)) which are computed for a parity-check matrix H of C. 6 The check is required only for k ∈ {2, 3}, as the remaining values of k are covered by Examples 3 and 4. The fact that the elements α j range over the quadratic residues of GF(11)-including zero-may or may not be coincidental. 7 Nonzero vectors that have disjoint supports are necessarily distinct. ...
An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters [n,k,d] and the list size L. L-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed-Solomon (GRS) codes.
... . On a F [2] d (n) 1, 5119n, et si g 3 F ...
... En ce qui concerne la minoration, d'autres constructions ont vu le jour depuis, comme celle de Ruzsa [28] ou celle de Bose [2] mais aucune n'a significativement amélioré le résultat. Si bien qu'aujourd'hui encore nous ne savons toujours pas si la différence F 1, n − √ n est non majorée, ni même si elle est positive. ...
... Soit α ∈ R. On appelle A α (A α = A α (n)) l'ensemble des unions de n/α 2 De plus cette dernière union est en fait disjointe car 2(i + 1)p − (2ip + n) = p + O(p 5/8 ) > 0 quel que soit i ∈ 2 F p (et p assez grand), donc 2ip + 1, 2ip + n − 1 ∩ 2jp + 1, 2jp + n − 1 = ∅ quels que soient i = j. Ainsi Le but est donc d'améliorer ces bornes afin de réussir à trouver la bonne asymptotique, si elle existe. ...
Our project lies in the field of additive combinatorics. More precisely, we seek the maximal size of a progression free subset of a finite group G, meaning a subset with no three distinct elements of the form a, a+d, a+2d (called a 3AP for 3 arithmetic progression). A 3AP is a simple and natural pattern that we expect to find in a 'large enough' set and we shall try to precise what 'large enough' means here. Trying to determine the maximal size of a progression free set is now a classical problem in additive combinatorics, on which many of the best experts have worked. There are two different aspects in this problem : to determine a minimal size for A which assures the existence of 3AP in A, this gives an upper bound for the maximal size of a progression free set; to build some large progression free sets, this gives a lower bound for this maximal size. We will insist on the constructive part in the context of groups Z_q^n with small q. We shall also try to adapt a construction by Ruzsa to this context. The progression of this work should be from some combinatorial constructions, allowing numerical approach, to more theoretical concepts.
... This yields a sequence of sets [1,2] ⊃ E 1 ⊃ E 2 ⊃ E 3 ⊃ . . . , where each E j consists of N j 0 intervals of length N −j . ...
... It is easy to see that µ j converge weakly as j → ∞ to a probability measure µ supported on the Cantor set E ∞ = ∞ j=1 E j , and that E ∞ has Hausdorff and Minkowski dimensions both equal to α := log N 0 log N (so that N 0 = N α ). Furthermore, there is a constant C µ > 0 such that for all x ∈ supp µ we have (1) C −1 µ r α ≤ µ((x − r, x + r)) ≤ C µ r α ∀ r > 0. ...
... To see this, let 1], and ϕ is supported in |ξ| ...
For , , and , we construct Cantor-type measures on supported on sets of Hausdorff dimension for which the associated maximal operator is bounded from to . Maximal theorems for fractal measures on the line were previously obtained by Laba and Pramanik [17]. The result here is weaker in that we are not able to obtain estimates; on the other hand, our approach allows Cantor measures that are self-similar, have arbitrarily low dimension , and have no Fourier decay. The proof is based on a decoupling inequality similar to that of Laba and Wang [18].
... Partial geometric designs come from many combinatorial and geometric structures (cf. [3,8,34]). In [45], by using an aid of the computer, van Dam and Spence classified and gave a complete list of small PGDs (having the sum of the numbers of points and blocks less than 36). ...
... An SPBIBD of type (s, t) is a PBIBD with two associate classes with the additional property that for any point-block pair (a, B) the number of first associates of a in the block B is s or t depending on whether a ∈ B or a / ∈ B, respectively (cf. [3] for PBIBD). ...
... Thus, we have a PGD with concurrence type [2, 2, 2] that is not a 2-design. This design is of type [4,4,3]. ...
We survey partial geometric designs and investigate their concurrences of points. It is well-known that the concurrence matrix of a partial geometric design can have at most three distinct eigenvalues, all of which are non-negative integers. The concurrence type of a design gives a simple profile of the incidence structure of the design. Namely, an ordinary 2- design has a single concurrence , and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design TD has two concurrences and 0, and its concurrence matrix is circulant. In this paper, we show the existence of other partial geometric designs having two or three concurrences, and investigate which symmetric circulant matrices are realized as the concurrence matrices of partial geometric designs. We collect some examples partial geometric designs and study their characteristics and construction methods. We give a list of partial geometric designs of order up to 12 each of which has a circulant concurrence matrix. We then classify these partial geometric designs along with their combinatorial properties and constructions.
... Let t ≥ 2 be an integer. There exists a constant n t such that, if θ min (G) ≥ −t, n ≥ n t and k < n − 2 − (t−1) 2 4 , then either G is the s-clique extension of a strongly regular graph for 2 ≤ s ≤ t − 1 or G is a p × q-grid with p > q ≥ 2. ...
... A parallel class S is a set of q mutually parallel planes in F 3 q . Bose first considered these sets in [2] to construct certain designs. In the next result we show that there are special parallel classes in F 3 q . ...
Tan et al. conjectured that connected co-edge-regular graphs with four distinct eigenvalues and fixed smallest eigenvalue, when having sufficiently large valency, belong to two different families of graphs. In this paper we construct two new infinite families of connected co-edge-regular graphs with four distinct eigenvalues and fixed smallest eigenvalue, thereby disproving their conjecture. Moreover, one of these constructions demonstrates that clique-extensions of Latin Square graphs are not determined by their spectrum.
... This theorem of Singer is also used to construct Golomb rulers and Sidon sets (For a survey, see Dimitromanolakis [10]), and in fact Montgomery's construction is in this setting equivalent to Ruzsa's construction (see [12] Theorem 2) which is also used to construct Golomb rulers and Sidon sets. There also exists a third construction of Bose [6] which is used to construct Golomb rulers that has hitherto not been used in the corresponding power sum problem. We will now see what this construction will yield when applied to the power sum problem. ...
... We first state a result taken from Bose [6]: ...
Let s_v denote the pure power sum \sum_{k=1}^n z_k^v. In a previous paper we proved that \sqrt n <= \inf_{|z_k| => 1} \max_{v=1,...,n^2} |s_v| <= \sqrt{n+1} when n+1 is prime. In this paper we prove that \inf_{|z_k| = 1} \max_{v=1,...,n^2-n} |s_v| = \sqrt{n-1} when n-1 is a prime power, and if 2 <= i <= n-1 and n => 3 is a prime power then \inf_{|z_k| => 1} \max_{v=1,...,n^2-i} |s_v| =\sqrt n. We give explicit constructions of n-tuples (z_1,...,z_n) which we prove are global minima for these problems. These are two of the few times in Turan power sum theory where solutions in the inf max problem can be explicitly calculated.
... These sets have a long history going back to Sidon [12] in the 30s and to the seminal work of Bose, Chowla and Singer as for lower bounds, and of Erdős and Turán as for upper bounds. See [13,1,2,4]. Except in the case g = 1, for which it is known that ...
... However, for any positive g, since a Sidon set is in particular a B 2 [g] set, it is known that the quantity F (g, N) grows at least like a constant times √ N. Better lower bounds were obtained in [11,7,3,9]. As for upper bounds, the current best result is (1) F (g, N) min 3.1694 g, 1.74217 (2g − 1) N , the first argument in the minimum being contained in [10] and the second one in [15]. Notice that the first is better than the second as soon as g ≥ 6. ...
Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 [g] sets and defined by the fact that there are at most g ways (up to reordering the summands) to represent a given integer as a sum of two elements of the set, are much more difficult to handle and not as well understood. In this article, using a numerical approach, we improve the best upper estimates on the size of a B 2 [g] set in an interval of integers in the cases g = 2, 3, 4 and 5.
... We first describe briefly the notions and tools we will need for the proof of Theorem 1. 4. In what follows, all logarithms are to the base e. ...
... The constructions of Singer [24] and Bose [4] yield affirmative answers to Question 4.3 when n is of the form q 2 + q + 1 or q 2 − 1 respectively, where q is a prime 13 power, and a construction due to Ruzsa [23] does so when n is of the form p 2 − p, where p is prime; as observed by Banakh and Gavrylkiv [1], these constructions of Singer, Bose and Ruzsa yield efficient difference covers as well, so we also have affirmative answers to Questions 4.2 and 1.6 for all n of the aforementioned form. ...
We make some progress on a question of Babai from the 1970s, namely: for with , what is the largest possible cardinality s(n,k) of an intersecting family of k-element subsets of admitting a transitive group of automorphisms? We give upper and lower bounds for s(n,k), and show in particular that as if and only if for some function that increases without bound, thereby determining the threshold at which `symmetric' intersecting families are negligibly small compared to the maximum-sized intersecting families. We also exhibit connections to some basic questions in group theory and additive number theory, and pose a number of problems.
... The structure of dense Sidon sets has a rich literature [11,29] and classic constructions by Erdős-Turán [13], Singer [33], Bose [2], Spence [14,30], Hughes [20] and Cilleruelo [6] have established that a dense Sidon set A satisfies |A| ≥ (1 − o(1)) √ n. As remarked by Ruzsa, "somehow all known constructions of dense Sidon sets involve the primes" [31]. ...
... Theorem B (R. C. Bose [2]). For a prime p, there are at least p elements in [p 2 − 1] such that the sums of two of these elements are different modulo p 2 − 1. ...
We prove that if is a Sidon set so that , then where . As an application of this, we give easy proofs of some previously derived results. We proceed on to proving that for any , we have for all but at most exceptions.
... In this work we use Bose-type Sidon sets to propose a new counter-based pseudorandom bit generator. We identify that Bose-type Sidon sets [19] (subsets of an additive set with the property that all sums of two elements of this set are different), denoted by B q (θ), satisfy the property that by reducing modulo q + 1 each element of B q (θ), where q is a prime power, generates, in an unpredictable way, the set of integers [1, q] =: {1, . . . , q} [20], which will represent the set of states of our pseudorandom bit generator. ...
... In this section we present the construction of our family of binary sequences that we call Binary Bose Sequences Generator or simply BB Sequences Generator, based on the construction of Sidon sets due to Bose [19]. ...
Sidon sets have a variety of applications in mathematics and in real-world problems, such as error-correcting codes, the signal compression in telecommunications, and cryptography. In particular, pseudorandom number (bit) generators with good randomness qualities is critical in the development of secure applications. Based on the construction of Bose-type Sidon sets, in this paper we present the construction of a new pseudorandom bit generator (PRBG) with large period, high degree of entropy and good randomness properties.
MSC Classification: 43A46 , 12E20 , 94A55
... The first published usage of the "B h " terminology that we have found is in the title of a paper of Mian & Chowla [8]. Perhaps the "B" is an homage to Bose, who had already produced a simple (yet thick) construction B 2 sets [3]. This work generalizes constructions of Bose-Chowla [4], itself a generalization of Bose's construction [3], and of Singer [13]. ...
... Perhaps the "B" is an homage to Bose, who had already produced a simple (yet thick) construction B 2 sets [3]. This work generalizes constructions of Bose-Chowla [4], itself a generalization of Bose's construction [3], and of Singer [13]. Definition 1. ...
A subset A of a commutative semigroup X is called a set in X if the only solutions to (with ) are the trivial solutions (as multisets). With h=2 and , these sets are also known as Sidon sets, Golomb Rulers, and Babcock sets. In this work, we generalize constructions of Bose-Chowla and Singer and give the resultant bounds on the diameter of a k element set in for small k. We conclude with a list of open problems.
... Group divisible t-designs are a class of critical important combinatorial designs proposed by Bose [4] in 1942. In particular, when the group size is 1, a group divisible t-design is a t-design. ...
... In particular, when λ = 1 t , it will be omitted. [2], [3], [4] ...
Cameron defined the concept of generalized ‐designs, which generalized ‐designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized ‐designs and group divisible ‐designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible ‐designs, and then point out close connections with various well‐known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible ‐designs and almost completely determine their existence for and small block sizes.
... A topic of particular interest is to determine the maximum size of a Sidon set contained in the first n positive integers. Seminal results of Erdős and Turán [5] concerning the upper bound, as well as Ruzsa [9], Bose [2] and Singer [11] for the lower bound established the following result. ...
... Theorem 1 ( [5,11,2,9]). A maximum size Sidon set A ⊂ [n] satisfies |A| = (1±o(1)) √ n. ...
A family of subsets of an abelian group G is a \emph{Sidon system} if the sumsets A+B with are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size of a Sidon system consisting of k-subsets of the first n positive integers satisfies for some constant only depending on k. We close the gap by proving an essentially tight structural result that in particular implies . We also use this to establish a result about the size of the largest Sidon system in the binomial random family . Extensions to h-fold sumsets for any fixed are also obtained.
... In the other direction, Chowla [5] and Erdős [9] independently found that the constructions of Singer [24] and Bose [4] provided the evidences of S n (1 + o(1))n 1/2 . ...
... Lemma 2.9 (Bose). [4] Let p be a prime, then there are at least p elements between [1, p 2 − 1] such that all the sums of two of these elements are different modulo p 2 − 1. ...
A set is called a Sidon set if all the sums are different. Let be the largest cardinality of the Sidon sets in . In a former article, the author proved the following asymptotic formula where is an arbitrary small constant. In this note, we give an extension of the above formula. We show that for any positive integers . Besides, we also consider the asymptotic formulae of other type summations involving Sidon sets. The proofs are established in a more general setting, namely we obtain the asymptotic formulae of the Sidon sets with t elements when t is near the magnitude .
... Bose [2] probó un análogo del Teorema de Singer, en 1942. Teorema 1.2. ...
... Como en el anillo de enteros módulo n tener diferencias no cero distintas es equivalente a tener sumas diferentes, los teoremas de Singer y Bose [13,2] se pueden enunciar en términos de sumas; es decir, se tienen las siguientes consecuencias. Corolario 1.3. ...
Un conjunto Bh es un subconjunto A de números enteros con la propiedad que todas las sumas de h elementos son distintas, salvo permutaciones de los sumandos. El problema fundamental consiste en determinar el máximo cardinal de un conjunto Bh contenido en el intervalo entero [1, n] := {1, 2, 3, . . . , n}. Se conocen pocas construcciones de conjuntos Bh enteros, entre ellas se tienen la de Singer [13], Bose-Chowla [3] y Gómez-Trujillo [7]. El concepto de conjunto Bh se puede extender a grupos arbitrarios. En este articulo se presentan las construcciones generalizadas a los grupos que provienen de un cuerpo y se obtiene una nueva construcción de un conjunto Bh+s en h + 1 dimensiones.
... We say that a Golomb ruler of order m is optimal if it has the shortest length possible (optimally short). For example, the ruler given in (1) where m = 15 is a optimal Golomb ruler with length 151 and G(15) = 151. Currently, there are also optimal Golomb rules where 2 ≤ m ≤ 27 marks [10], [12] and there is an ongoing search for an optimal 28-marks rule. ...
... For all q prime power, the Bose construction provides a Golomb ruler with q marks and modulus q 2 − 1, [1]. Using a suitable group homomorphism and the Lemma 2 we have the following result: ...
A set of positive integers A is called a
g
-Golomb ruler if the difference between two distinct elements of A is repeated up to
g
times. This definition is a generalization of the Golomb ruler
. In this paper, we obtain new constructions for
g
-Golomb rulers from Golomb rulers, using these constructions we find some suboptimal 2 and 3-Golomb rulers with up to 124 marks and we prove two theorems related to extremal functions associated with this sets improving already known results.
... Thus for any U 1 , U 2 , U 3 , U 4 ⊂ S satisfying U i ∩ U j = ∅ for i = j, the set U 1 ∪ (U 2 + p(p − 1)) ∪ (U 3 + 2p(p − 1)) ∪ (U 4 + 3p(p − 1)) is a Sidon subset of [4p(p − 1)], where V + x := {v + x : v ∈ V }. This gives (1)) √ n > 2 (1.16+o (1) , where p is the largest prime such that 4p(p − 1) < n, and using the fact that the ratio of successive primes tends to 1. (We note that any construction for large modular Sidon sets could have been used here; this includes the classical constructions of Singer [44] and of Bose [8].) ...
A set of containers for a hypergraph G is a collection of vertex subsets, such that for every independent (or, indeed, merely sparse) set in G there is some subset in the collection which contains it. No set in the collection should be large and the collection itself should be relatively small. Containers with useful properties have been exhibited by Balogh, Morris and Samotij and by the authors, along with several applications. Our purpose here is to give a simpler algorithm than the one we used previously, which nevertheless yields containers with all the properties needed for our previous theorem. Moreover this algorithm produces containers having the so-called online property, allowing previous colouring applications to be extended to all, not just simple, hypergraphs. For illustrative purposes, we include a complete proof of a slightly weaker but simpler version of the theorem, which for many (perhaps most) applications is plenty. We also present applications to the number of solution-free sets of linear equations, including the number of Sidon sets, announced previously but not proven.
... None the less, we can always determine a smaller code which is cyclic and still related with our geometries. To this purpose, we shall make use of the affine Singer cyclic group S of PG (r − 1, q t ); see [4,20]. This is a linear collineation group which has exactly 3 orbits in PG (r − 1, q t ): an hyperplane Σ, a single point O with O ∈ Σ and the points of AG(r − 1, q t ) = PG (r − 1, q t ) \ Σ different from O. The action on the latter orbit is regular. ...
Using geometric properties of the variety \cV_{r,t}, the image under the Grassmannian map of a Desarguesian -spread of \PG(rt-1,q), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We exactly determine the parameters of these codes and characterise the words of minimum weight.
... Also note that if D is a B 2sequence over Z n and a ∈ Z n , then so is the shift a + D = {a + d : d ∈ D}. The following theorem, due to Bose [43], shows that large B 2 -sequences over Z n exist for many values of n. Theorem 23. ...
A two-dimensional grid with dots is called a \emph{configuration with distinct differences} if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various two-dimensional shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid.
... The following theorem was derived in [2] from the classical results of Singer [13], Bose, Chowla [4], [5] and Rusza [12]. ...
A subset B of a group G is called a difference basis of G if each element can be written as the difference of some elements . The smallest cardinality of a difference basis is called the difference size of G and is denoted by . The fraction is called the difference characteristic of G. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number , any finite Abelian p-group G has difference characteristic . Also we calculate the difference sizes of all Abelian groups of cardinality .
... The following theorem was derived in [2] from the classical results of Singer [11], Bose, Chowla [3], [4] and Rusza [10]. ...
A subset B of a group G is called a difference basis of G if each element can be written as the difference of some elements . The smallest cardinality of a difference basis is called the difference size of G and is denoted by . The fraction is called the difference characteristic of G. We prove that for every the dihedral group of order 2n has the difference characteristic . Moreover, if , then . Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality .
... Los enteros son radio frecuencias y como es deseable tener un rango espectral pequeño, se buscan soluciones en las que a m sea lo mínimo posible (solucionesóptimas) o al menos probablemente cercanas a seróptimas (soluciones subóptimas)". Utilizando los conjuntos con diferencias distintas obtenidos por James Singer, (Singer, 1938) y un resultado análogo (Bose, 1942), en (Atkinson, Santoro, & Urrutia, 1986), se demuestra que: ...
Un conjunto A de enteros se llama conjunto Sidon si todas las sumas de dos elementos en A son distintas; es decir si para todo a, b, c, d ∈ A (a + b = c + d)⇒ {a,b} = {c,d}. Estos conjuntos los consideró el analista Simon Sidon, a principios de los años 1930, en sus investigaciones sobre análisis de Fourier. Una regla Golomb es un conjunto de enteros (no negativos) con la propiedad que todas las diferencias no cero son distintas. Estas reglas especiales aparecieron en el estudio de interferencias en radiofrecuencias, realizado por Wallace Babcock, a principios de los años 1950. En este artículo se realiza un recorrido a través de contextos finitos en los que intervienen estos objetos matemáticos. Los contextos considerados son: conjuntos finitos de números enteros, grupos cíclicos, retículos bidimensionales de coordenadas enteras (arreglos Costas, secuencias Sonar) y funciones APN. En buena parte, estas notas son resultado parcial de actividades realizadas durante mi actual año sabático en la Universidad del Cauca, y son también una versión ampliada de la conferencia invitada, que con el mismo título presenté durante el “55 Congreso Nacional de la Sociedad Matemática Mexicana”, realizado en la Universidad de Guadalajara del 23 al 28 de octubre de 2022 (https://www.smm.org.mx/congreso).
... We will introduce the Bose-Chowla construction of Sidon sets [BC60,Bos42]. ...
We present an explicit construction of a sequence of rate 1/2 Wozencraft ensemble codes (over any fixed finite field ) that achieve minimum distance where k is the message length. The coefficients of the Wozencraft ensemble codes are constructed using Sidon Sets and the cyclic structure of where k+1 is prime with q a primitive root modulo k+1. Assuming Artin's conjecture, there are infinitely many such k for any prime power q.
... Construction 3 (Bose [Bos42]). Let K be a finite field and let L be an extension of K of degree 2. Let G = L × . ...
The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples. There are many further examples arising from nondesarguesian planes. We conjecture that all dense Sidon sets arise from finite projective planes in this way. If true, this implies that all abelian groups of most orders do not have dense Sidon subsets. In particular if denotes the size of the largest Sidon subset of , this implies . We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.
... A topic of particular interest is to determine the maximum size of a Sidon set contained in the first n positive integers. Seminal results of Erdős and Turán [5] concerning the upper bound, as well as Ruzsa [10], Bose [2] and Singer [12] for the lower bound established the following result. Theorem 1 ([5, 12, 2, 10]). ...
A family of subsets of an abelian group G is a Sidon system if the sumsets A+B with are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size of a Sidon system consisting of k-subsets of the first n positive integers satisfies for some constant only depending on k. We close the gap by proving an essentially tight structural result that in particular implies . We also use this to establish a result about the size of the largest Sidon system in the binomial random family . Extensions to h-fold sumsets for any fixed are also obtained.
... In 1939, Singer [10] gave a construction of Sidon sets that, in practice, is quite excellent. In 1942, Bose [2] gave another. Shearer [8] and more recently Dogon-Rokicki [4] have taken this computation to its logical extreme. ...
A Sidon set is a set of integers containing no nontrivial solutions to the equation a+b=c+d. We improve on the lower bound on the diameter of a Sidon set with k elements: if k is sufficiently large and is a Sidon set with k elements, then . Alternatively, if n is sufficiently large, then the largest subset of that is a Sidon set has cardinality at most . While these are slight numerical improvements on Balogh-F\"uredi-Roy (arXiv:2103:15850v2), we use a method that is logically simpler than theirs but more involved computationally.
... Pasaron algunos años hasta que esta construcción fuera conocida por Erdős y popularizada en el mundo de los conjuntos de Sidon. Posteriormente Bose (1942) construyeron un conjunto de Sidon de q elementos en Z q 2 1 . Estas construcciones, los Ejemplos 1, 2, 3 y la cota superior en (11) prueban el valor exacto de F 2 (G) para algunas familias de grupos: ...
These are lecture notes for the AGRA II school, which took place in August 2015 at Universidad de San Antonio Abad del Cusco (Per\'u). They are geared towards graduate students and young researchers. I. Modular forms and Shimura curves (R. J. Miatello, G. Tornar\'ia, A. Pacetti, M. Harris) II. Additive combinatorics (J. Ru\'e, J. Wolf, J. Cilleruelo, P. Candela) III. Introduction to elliptic curves (M. Rebolledo and M. Hindry) IV. Growth in groups and expanders (H. A. Helfgott, M. Belolipetsky) V. Analysis and geometry in groups (A. Zuk) VI. Equidistribution and Diophantine analysis (R. Menares)
... One objective of this work is to present new constructions of extended sonar sequences, these constructions use special one-dimensional Sidon sets as provided by Bose [25] and Ruzsa [26]. ...
A (m, n) sonar sequence is an m × n array with exactly one dot in each column and where all lines connecting two dots in the array are distinct as vectors. These arrays are known to have many applications such as sonar and radar detection and these are studied as a particular case of Golomb rectangles or two-dimensional Sidon sets. The main open problem for sonar sequences is: for fixed m, find the largest n for which there is an (m, n) sonar sequence, these sequences are called the best sonar sequences. The extended sonar sequences are generalizations of sonar sequences where each column has at most one dot, the motivation to study these arrays are the best results obtained when applied to radar and sonar detection. In this paper, we give the best sonar sequences with m ≤ 100 obtained from an exhaustive computational search based on the Caicedo, Ruiz and Trujillo constructions and we present new constructions of extended sonar sequences that use Sidon sets.
... The bounds for S(n) have been progressively improved [ET41,Lin69,Cil10], the best current upper bound was given by Balogh, Füredi and Roy [BFR21] who established that S(n) ≤ n 1/2 + cn 1/4 , where c ≤ 0.998. It is also known that S(n) ≥ n 1/2 (1 − o(1)), which was proved independently by Singer [Sin38] and Bose [Bos42]. To obtain these last bounds, Singer and Bose work with X = G = Z n where they construct Sidon sets algebraically. ...
Given a positive integer k, the Sidon-Ramsey number SR(k) is defined as the minimum n such that in every partition of [1,n] into k parts there is a part containing two pairs of numbers with the same sum. In this note we show that . Along the way, we prove a variant of the Symmetric Hypergraph theorem which may be of independent interest. We also find the correct asymptotic behavior of an analogous problem in which the interval [1,n] is replaced with a d-dimensional box.
... Bose and Chowla [2,3] showed that A q is a q-element Sidon set in Z q 2 −1 , i.e., the numbers {a − a ′ : a, a ′ ∈ A, a = a ′ } are all distinct mod (q 2 − 1). The properties of the set A q can be found in many places, e.g., in Chapter 27 of the excellent textbook of van Lint and Wilson [16]. ...
Combining two elementary proofs we decrease the gap between the upper and lower bounds by in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of is at most for sufficiently large n.
... An affine difference set D gives rise to an affine plane (and hence to a projective plane Π q (D)) as follows: Points of the plane are the elements of G, together with a special point O (the origin), lines through O are the cosets of N, the remaining lines are of the form Dg, g ∈ G. We refer the reader to [5,11] and [10] for further information about affine difference sets. ...
In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph H, the maximum number k for which there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane PG(2,q) for all q. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.
... It is widely conjectured that a (q 2 + q + 1, q + 1, 1)-difference set exists only if q is a prime power, and this conjecture has been verified for all q < 2, 000, 000; see [14]. Theorem 1.3 (Bose [3]). For any prime power q, there is a (q 2 − 1, q)-MGR. ...
In this paper, we propose a simple construction for binary (
n
,
k
) linear codes using s-mark Golomb rulers. We prove that these codes are sequential-recovery locally repairable codes (LRCs) with availability 2, which can sequentially recover 5 erased symbols. We prove the necessary and sufficient condition for the proposed codes to be rate-optimal. We also prove the necessary and sufficient condition for the proposed codes to be dimension-optimal. Finally, we propose some variations of this constructions to obtain some 5-sequential recovery LRCs with availability 3. The proposed codes have higher rates and have more flexible choice for the lengths than other previously reported constructions.
Several new constructions of 3-dimensional optical orthogonal codes are presented here. In each case the codes have ideal autocorrelation , and in all but one case a cross correlation of . All codes produced are optimal with respect to the applicable Johnson bound either presented or developed here. Thus, on one hand the codes are as large as possible, and on the other, the bound(s) are shown to be tight. All codes are constructed by using a particular automorphism (a Singer cycle) of , the finite projective geometry of dimension k over the field of order , or by using an affine analogue in AG(k,q) .
We present an explicit construction of a sequence of rate 1/2 Wozencraft ensemble codes (over any fixed prime field F
q
) that achieve minimum distance Ω(√
k
) where
k
is the message length. The coefficients of the Wozencraft ensemble codes are constructed using Sidon Sets and the cyclic structure of F
q<sup>k</sup>
where
k
+ 1 is prime with
q
a primitive root modulo
k
+ 1. Assuming Artin’s conjecture, there are infinitely many such
k
for any prime
q
.
Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields. We present the first noncyclotomic infinite families of DPDFs which are also EPDFs, in structures other than finite fields (in particular cyclic groups and nonabelian groups). As well as direct constructions, we present an approach to constructing DPDFs/EPDFs using relative difference sets (RDSs); as part of this, we demonstrate how the well‐known RDS result of Bose extends to a very natural construction for DPDFs and EPDFs.
Let be a free group of rank n, with free generating set X. A subset D of is a \emph{Distinct Difference Configuration} if the differences are distinct, where g and h range over all (ordered) pairs of distinct elements of D. The subset D has diameter at most d if these differences all have length at most d. When n is fixed and d is large, the paper shows that the largest distinct difference configuration in of diameter at most d has size approximately .
For any n and k, we provide an explicit (that is, computable in polynomial time) example of integer -sequence of size n consisting of elements bounded by .
A Sidon set is a subset of an Abelian group with the property that the sums of two distinct elements are distinct. We relate the Sidon sets constructed by Bose to affine subspaces of of dimension one. We define Sidon arrays which are combinatorial objects giving a partition of the group as a union of Sidon sets. We also use linear recurring sequences to quickly obtain Bose-type Sidon sets without the need to use the discrete logarithm.
An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters
[n,k,d]
and the list size
L
.
L
-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed–Solomon (GRS) codes.
We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incidence structure of the design. An ordinary 2‐ ( v , k , λ ) design has a single concurrence λ , and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design TD λ ( k , u ) has two concurrences λ and 0, and its concurrence matrix is circulant. In this paper, we survey the known PGDs by highlighting their concurrences and constructions. Then we investigate which symmetric circulant matrices are realized as the concurrence matrices of PGDs. In particular, we try to give a list of all PGDs of order up to 12 each of which has a circulant concurrence matrix. We then describe these designs along with their combinatorial properties and constructions. This work is part of the second author's Ph.D. dissertation [46].
In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as , such that each represents the lines of a copy of the Fano plane . We generalize this observation by constructing, for each prime power q, a simplicial complex with vertices and facets consisting of two copies of . Our construction works for any colored k-configuration, defined as a k-configuration whose associated bipartite graph G is connected and has a k-edge coloring , such that for all , , following edges of colors from v brings us back to v. We give constructions of colored k-configurations from planar difference sets and commutative semifields. Then we give one-to-one correspondences between (1) Sidon sets of order 2 and size in groups with order n, (2) linear codes with radius 1 and index n in , and (3) colored -configurations with n points and n lines.
The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples, but there are many further examples arising from nondesarguesian planes. We conjecture that all dense Sidon sets arise in this manner. We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.
In his pioneering work on LDPC codes, Gallager dismissed codes with parity-check matrices of weight two after proving that their minimum Hamming distances grow at most logarithmically with their code lengths. In spite of their poor minimum Hamming distances, it is shown that quasi-cyclic LDPC codes with parity-check matrices of column weight two have good capability to correct phased bursts of erasures which may not be surpassed by using quasi-cyclic LDPC codes with parity-check matrices of column weight three or more. By modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. Quasi-cyclic LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel are also considered. Examples of such codes based on Reed-Solomon and Gabidulin codes are presented.
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