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Abstract: Abstract We revisit the seminal Brill–Noether algorithm for plane curves with ordinary singularities. Our new approach takes advantage of fast algorithms for polynomials and structured matrices. We design a new probabilistic algorithm of Las Vegas type that computes a Riemann–Roch space in expected sub-quadratic time. PubDate: 2022-12-01
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Abstract: Abstract Linear codes with few weights have applications in cryptography, association schemes, designs, strongly regular graphs, finite group theory, finite geometries, and secret sharing schemes, among other disciplines. Two-weight linear codes are particularly interesting because they are closely related to objects in different areas of mathematics such as strongly regular graphs, 3-rank permutation groups, ovals, and arcs. There exist techniques to construct linear codes with few weights, for example, the systematic exposition by Calderbank and Kantor (Bull Lond Math Soc 18(2):97–122, 1986). Ding et al., (World Sci, pp 119–124, 2008) and (IEEE Trans Inf Theory 61(11):5835–5842, 2015) constructed few-weight codes using the trace function and Tonchev et al. (Algorithms, 12(8), 2019) generalized Ding’s construction. In this paper, we present an elementary way to get two- and three-weight codes from simplex codes and antipodal linear codes. An interesting application is the construction of uniformly packed linear codes from two-weight codes and quaternary quasi-perfect linear codes from three-weight codes. PubDate: 2022-12-01
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Abstract: Abstract Complementary decompositions of monomial ideals—also known as Stanley decompositions—play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised algorithm. We show that Hironaka’s construction terminates, if and only if the monomial ideal is quasi-stable. Furthermore, we show that in this case the algorithm of Janet determines the same decomposition more efficiently. Finally, we briefly discuss how these results can be used for the computation of primary and irreducible decompositions. PubDate: 2022-12-01
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Abstract: Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers \(x=1, y=3\) is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if \(p<41\) is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then \(p\le 5\) . The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism \(\phi \) of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under \(\phi \) . PubDate: 2022-12-01
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Abstract: Abstract This paper is a contribution to the classification of parallelisms in three-dimensional projective spaces over small finite fields of order q by computer. The smallest space in which parallelisms have not yet been classified is for \(q=4.\) Partial results are available. The parallelisms admitting a nontrivial automorphism of odd prime order are known. Moreover, much is known about the case of parallelisms of \({{\mathrm{PG}}}(3,4)\) whose automorphism group is a two group. Namely, everything is known for two of the three possible groups of order two, as well as for cyclic groups of order 4. The present paper will settle the case of parallelisms whose automorphism group is elementary abelian of order 4. This leaves open the cases of parallelisms whose full automorphism groups are either trivial or a specific group of order two. PubDate: 2022-12-01
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Abstract: Abstract The Grothendieck ring of Chow motives admits two natural opposite λ-ring structures, one of which is a special structure allowing the definition of Adams operations on the ring. In this work I present algorithms which allow an effective simplification of expressions that involve both λ-ring structures, as well as Adams operations. In particular, these algorithms allow the symbolic simplification of algebraic expressions in the sub-λ-ring of motives generated by a finite set of curves into polynomial expressions in a small set of motivic generators. As a consequence, the explicit computation of motives of some moduli spaces is performed, allowing the computational verification of some conjectural formulas for these spaces. PubDate: 2022-12-01
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Abstract: Abstract The classification of unitals with parameters 2-(28, 4, 1) according to the 2-rank of their incidence matrices was initiated by McGuire, Tonchev and Ward, who proved that the 2-rank of any unital on 28 points is greater than or equal to 19, and up to isomorphism, there is a unique unital with 2-rank equal to 19. Jaffe and Tonchev investigated the next two 2-ranks, 20 and 21, and showed that there are no unitals on 28 points with 2-rank equal to 20, and there are exactly 4 isomorphism classes of unitals of rank 21. The subject of this paper is the classification of unitals having 2-rank 22, 23 and 24. PubDate: 2022-12-01
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Abstract: Abstract We describe software for symbolic computations that we developed in order to find Hamiltonian operators for Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, and verify their compatibility. The computation involves nonlocal (integro-differential) operators, for which specific canonical forms and algorithms have been used. PubDate: 2022-12-01
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Abstract: Abstract Moved by a question posed us by Wolfgang Rump, we investigate the Rump ideal \({\mathbb {I}}(p^2-pq+qp)\subset {\mathbb {Z}}\langle q,q^{-1}, p\rangle \) and we show, this way, the power of Zacharias representation. PubDate: 2022-11-07
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Abstract: Abstract Computer Algebra relies heavily on the computation of Gröbner bases, and these computations are primarily performed by means of Buchberger’s algorithm. In this overview paper, we focus on methods avoiding the computational intensity associated to Buchberger’s algorithm and, in most cases, even avoiding the concept of Gröbner bases, in favour of methods relying on linear algebra and combinatorics. PubDate: 2022-11-05
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Abstract: Abstract Let r be a divisor of \(q-1.\) An element \(\alpha \in {\mathbb {F}}_{q}\) is said to be r-primitive if ord \((\alpha )=\frac{q-1}{r}\) . In this paper, we discuss the existence of r-primitive pairs \((\alpha , f(\alpha ))\) where \(\alpha \in {\mathbb {F}}_q\) , f(x) is a general rational function of degree sum m (degree sum is the sum of the degrees of numerator and denominator of f(x)) and the denominator of f(x) is square-free. Then we show that for any integer \(m>0\) , there exists a positive constant \(B_{r,m}\) such that if \(q>B_{r,m}\) , then such r-primitive pairs exist. In particular, we present a bound for \(B_{r,m}\) with \(r=2\) and \(m\in \{2,3,4,5,6\}\) , and provide some conditions on the existence of 2-primitive pairs. PubDate: 2022-11-01
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Abstract: Abstract We propose a general greedy algorithm for binary de Bruijn sequences, called Generalized Prefer-Opposite Algorithm, and its modifications. By identifying specific feedback functions and initial states, we demonstrate that most previously-known greedy algorithms that generate binary de Bruijn sequences are particular cases of our algorithm. PubDate: 2022-11-01
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Abstract: Abstract We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group \({\mathbb {Z}}_{p^e}\) . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with \(p \equiv 1 \pmod {4}\) and e is odd with \(e>1\) ; and all lengths that are \(0 \pmod {4}\) when p is an odd prime with \(p \equiv 3 \pmod {4}\) and e is odd with \(e>1.\) PubDate: 2022-11-01
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Abstract: Abstract A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters \((n,\varepsilon ,q)\) is given in terms of Chebyshev polynomials. According to the parity of n and the sign \(\varepsilon\) , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or \(\varepsilon =-1\) using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture. PubDate: 2022-11-01
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Abstract: Abstract Traditional global stability measure for sequences is hard to determine because of large search space. We propose the k-error linear complexity with a zone restriction for measuring the local stability of sequences. For several classes of sequences, we demonstrate that the k-error linear complexity is identical to the k-error linear complexity within a zone, while the length of a zone is much smaller than the whole period when the k-error linear complexity is large. These sequences have periods \(2^n\) , or \(2^v r\) (r odd prime and 2 is primitive modulo r), or \(2^v p_1^{s_1} \cdots p_n^{s_n}\) ( \(p_i\) is an odd prime and 2 is primitive modulo \(p_i^2\) , where \(1\le i \le n\) ) respectively. In particular, we completely determine the spectrum of 1-error linear complexity with any zone length for an arbitrary \(2^n\) -periodic binary sequence. PubDate: 2022-11-01
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Abstract: Abstract In this paper, we consider smooth cubic surfaces with 15 lines. It is known that such surfaces can be generated by means of a double six with two pairs of Galois conjugate lines defined over the quadratic extension. The approach taken here is to consider the generation by means of a set of 9 lines defined over the field of coordinates. Eight lines arise from the double six, while the ninth is the diagonal line of the two pairs of Galois conjugate lines. This allows us to express all necessary equations and objects in terms of a set of four parameters over the coordinate field. As an application, we classify the smooth cubic surfaces with 15 lines over small finite fields by computer. All our results match with an enumerative formula recently found by Das. PubDate: 2022-10-03
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Abstract: Abstract We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in \(\mathbb {K}[x]\) . The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type \(f(\alpha )=f(\beta )\) for \(\alpha ,\beta\) in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras. PubDate: 2022-08-20 DOI: 10.1007/s00200-022-00573-4
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Abstract: Abstract We consider Noetherian operators in the context of symbolic computation. Upon utilizing the theory of holonomic \({\mathcal D}\) -modules, we present a new method for computing Noetherian operators associated to a zero-dimensional ideal. An effective algorithm that consists mainly of linear algebra techniques is proposed for computing them. Moreover, as applications, new computation methods of polynomial ideals are discussed by utilizing the Noetherian operators. PubDate: 2022-08-08 DOI: 10.1007/s00200-022-00570-7
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