In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod ... more In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod 3). We also show that 4-GDDs of type 2t8s exist for all but a finite number of feasible values of s and t. The largest unknown case has type 24818 and has 152 points. A number of 4-GDDs with at most 50 points are also constructed. These include one of type 4811101, the last feasible type of the form 4s1tn1 with at most 50 points for which no 4-GDD was known
ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear co... more ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.
ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assm... more ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assmus-Mattson theorem and another coding-theoretical theorem with respect to matroids, and thereby present new sufficient conditions for obtaining $t$-designs from matroids. These conditions may be relaxed for some self-dual matroids. We use our results to prove new constructions of $t$-designs from linear codes, including several new extensions and variants of the Assmus-Mattson theorem. We also present weighted $t$-designs which generalize $t$-designs. New $t$-designs are obtained from our results.
In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod ... more In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod 3). We also show that 4-GDDs of type 2t8s exist for all but a finite number of feasible values of s and t. The largest unknown case has type 24818 and has 152 points. A number of 4-GDDs with at most 50 points are also constructed. These include one of type 4811101, the last feasible type of the form 4s1tn1 with at most 50 points for which no 4-GDD was known
ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear co... more ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.
ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assm... more ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assmus-Mattson theorem and another coding-theoretical theorem with respect to matroids, and thereby present new sufficient conditions for obtaining $t$-designs from matroids. These conditions may be relaxed for some self-dual matroids. We use our results to prove new constructions of $t$-designs from linear codes, including several new extensions and variants of the Assmus-Mattson theorem. We also present weighted $t$-designs which generalize $t$-designs. New $t$-designs are obtained from our results.
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Papers by Thomas Britz