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Since its establishment in 1974, Aalborg University has aimed to deliver cutting-edge research and innovative practices in engineering education. However, increasing complexity related to rapid developments in digitalization and emerging societal challenges offer new opportunities, requirements, and expectations across disciplines for educational institutions and staff to develop and adapt to an unpredictable and ever-changing future. This paper proposes a systemic approach to addressing these challenges in higher education and presents recent initiatives launched at AAU including a coming Institute of Advanced Study in PBL to support the digital transformation of problem- and project-based learning for the future.

Since its establishment in 1974, Aalborg University has aimed to deliver cutting-edge research and innovative practices in engineering education. However, increasing complexity related to rapid developments in digitalization and emerging societal challenges offer new opportunities, requirements, and expectations across disciplines for educational institutions and staff to develop and adapt to an unpredictable and ever-changing future. This paper proposes a systemic approach to addressing these challenges in higher education and presents recent initiatives launched at AAU including a coming Institute of Advanced Study in PBL to support the digital transformation of problem- and project-based learning for the future. [ABSTRACT FROM AUTHOR]

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout's theorem, IEEE Trans. Inform. Theory46(2) (2000) 635–641, On (or in) Dick Blahut's 'footprint', Codes , Curves Signals (1998) 3–9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class. [ABSTRACT FROM AUTHOR]

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield subcodes of $J$ -affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield subcodes of $J$ -affine variety codes and provide some new and good LCD codes coming from this construction. [ABSTRACT FROM AUTHOR]

In two recent papers the first by Feng, Rao, Berg, and Zhu and the second by Feng, Zhu, Shi, and Rao, the authors use a generalization of Bezout's theorem to estimate the minimum distance and generalized Hamming weights for a class of error-correcting codes obtained by evaluation of polynomials in points of an algebraic curve. The main aim of this note is to show that instead of using this rather complex method the same results and some improvements can be obtained by using the so-called footprint from Grobner basis theory. We also develop the theory further such that the minimum distance and the generalized Hamming weights not only can be estimated but also can actually be determined. Index Terms--Evaluation codes, generalized Hamming weights, minimum distance.

Two new constructions of linear code pairs C2 \subset C1 are given for which the codimension and the relative minimum distances M1(C1,C2) and M1(C\vphantom {Rj2}^\perp,C1^\perp) are good. By this, we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes. They can also be used to ensure reliable communication over asymmetric quantum channels. The new constructions result from carefully applying the Feng–Rao bounds to a family of codes defined from multivariate polynomials and Cartesian product point sets. [ABSTRACT FROM AUTHOR]

Coset constructions of q-ary Reed-Muller codes can be used to store secrets on a distributed storage system in such a way that only parties with access to a large part of the system can obtain information while still allowing for local error-correction. In this paper we determine the relative generalized Hamming weights of these codes which can be translated into a detailed description of the information leakage [2, 21, 18, 9]. [ABSTRACT FROM AUTHOR]

We consider multivariate polynomials and investigate how many zeros of multiplicity at least r they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of multiplicity that we use is the one related to Hasse derivatives. As a generalization of material in Augot and Stepanov (2009) and Dvir et al. (2009) a general version of the Schwartz–Zippel was presented in Geil and Thomsen (2013) which from the leading monomial – with respect to a lexicographic ordering – estimates the sum of zeros when counted with multiplicity. The corresponding corollary on the number of zeros of multiplicity at least r is in general not sharp and therefore in Geil and Thomsen (2013) a recursively defined function D was introduced using which one can derive improved information. The recursive function being rather complicated, the only known closed formula consequences of it are for the case of two variables (Geil and Thomsen, 2013). In the present paper we derive closed formula consequences for arbitrary many variables, but for the powers in the leading monomial being not too large. Our bound can be viewed as a generalization of the footprint bound (Høholdt, 1998; Geil and Høholdt, 2000)—the classical footprint bound taking not multiplicity into account. [ABSTRACT FROM AUTHOR]

Asymptotically good sequences of ramp secret sharing schemes were given in [5] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. Their security is given by the relative generalized Hamming weights of the corresponding codes. In this paper we demonstrate how to obtain refined information on the RGHWs when the codimension of the codes is small. For general codimension, we give an improved estimate for the highest RGHW. [ABSTRACT FROM AUTHOR]

Kopparty and Wang studied in [3] the relation between the roots of a univariate polynomial over F q and the zero–nonzero pattern of its coefficients. We generalize their results to polynomials in more variables. [ABSTRACT FROM AUTHOR]

Salazar, Dunn and Graham in [16] presented an improved Feng-Rao bound for the minimum distance of dual codes. In this work we take the improvement a step further. Both the original bound by Salazar et al., as well as our improvement are lifted so that they deal with generalized Hamming weights. We also demonstrate the advantage of working with one-way well-behaving pairs rather than weakly well-behaving or well-behaving pairs. [ABSTRACT FROM AUTHOR]

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper, we elaborate on the implication of these parameters and devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes, our bound is often tight. Furthermore, for these codes, the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights. [ABSTRACT FROM AUTHOR]

Abstract: We show that the Feng–Rao bound for dual codes and a similar bound by Andersen and Geil (2008) [1] for primary codes are consequences of each other. This implies that the Feng–Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura (2000) [30] (see also Beelen and Høholdt, 2008 [3]) derived from the Feng–Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition in Matsumoto and Miura (2000) [30] requires the use of differentials which was not needed in Andersen and Geil (2008) [1]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miuraʼs bound and Andersen and Geilʼs bound when applied to primary one-point algebraic geometric codes. [Copyright &y& Elsevier]

Abstract: The celebrated Feng–Rao bound estimates the minimum distance of codes defined by means of their parity check matrices. From the Feng–Rao bound it is clear how to improve a large family of codes by leaving out certain rows in their parity check matrices. In this paper we derive a simple lower bound on the minimum distance of codes defined by means of their generator matrices. From our bound it is clear how to improve a large family of codes by adding certain rows to their generator matrices. The new bound is very much related to the Feng–Rao bound as well as to Shibuya and Sakaniwa''s bound in [T. Shibuya, K. Sakaniwa, A dual of well-behaving type designed minimum distance, IEICE Trans. Fund. E84-A (2001) 647–652]. Our bound is easily extended to deal with any generalized Hamming weights. We interpret our methods into the setting of order domain theory. In this way we fill in an obvious gap in the theory of order domains. [Copyright &y& Elsevier]

The main results of this paper are derived by using only simple Gro¨bner basis techniques. We present a new construction of evaluation codes from Miura–Kamiya curves Cab. We estimate the minimum distance of the codes and estimate the minimum distance of a class of related one-point geometric Goppa codes. With respect to these estimates the new codes perform at least as well as the related geometric Goppa codes. In particular we consider codes from norm–trace curves. We show that our estimates give actually the true minimum distance of these codes. The new codes from norm–trace curves perform rather well. In many cases much better than the corresponding geometric Goppa codes. It turns out that an alternative description of the new codes from norm–trace curves can be made by using Høholdt et al.'s in: V.S. Pless, W.C. Huffman (Eds.), Handbook of Coding Theory, Vol. 1, Elsevier, Amsterdam, 1998, pp. 871–961 (Chapter 10) construction of improved dual codes. [Copyright &y& Elsevier]

The notion of an order domain is generalized. The behaviour of an order domain by taking a subalgebra, the extension of scalars, and the tensor product is studied. The relation of an order domain with valuation theory, Gro¨bner algebras, and graded structures is given. The theory of Gro¨bner bases for order domains is developed and used to show that the factor ring theorem and its converse, the presentation theorem, hold. The dimension of an order domain is related to the rank of its value semigroup. [Copyright &y& Elsevier]

Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [15] and in the CSS construction of quantum codes [14]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with these values and with the remaining parameter being as large as possible. In this work we study nested codes from the Hermitian curve. For not too small codimension, we present improved constructions and provide closed formula estimates on their performance. For small codimension we show how to choose pairs of one-point algebraic geometric codes in such a way that one of the relative minimum distances is larger than the corresponding non-relative minimum distance. [ABSTRACT FROM AUTHOR]