Generalized Steiner triple systems, GS(2, 3, n, g) are equivalent to (g+1)-ary maximum constant weight codes (n, 3,3)s. In this paper, it is proved that the necessary conditions for the existence of a GS(2,3, n, 10), ...Generalized Steiner triple systems, GS(2, 3, n, g) are equivalent to (g+1)-ary maximum constant weight codes (n, 3,3)s. In this paper, it is proved that the necessary conditions for the existence of a GS(2,3, n, 10), namely, n ≡ 0,1 (mod 3) and n ≥ 12, are also sufficient.展开更多
A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposab...A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposable and denoted by IDLSTSx（v） if there does not exist an LSTSx, （v） contained in the collection for any λ 〈 λ. In this paper, we show that for λ = 5, 6, there is an IDLSTSλ（v） for v ≡ 1 or 3 （rood 6） with the exception IDLSTS6（7）.展开更多
Generalized Steirier triple systems, GS(2,3,n,g), are equivalent to maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The necessary conditions...Generalized Steirier triple systems, GS(2,3,n,g), are equivalent to maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The necessary conditions for the existence of a GS(2,3,n,g) are (n-1)g≡0 (mod 2), n(n-1)g2≡0 (mod 6), and n≥g+2. These necessary conditions are shown to be sufficient by several authors for 2≤g≤11. In this paper, three new results are obtained. First, it is shown that for any given g, g≡0 (mod 6) and g≥12, if there exists a GS(2.3.n.g) for all n, g+2≤n≤7g+13. then the necessary conditions are also sufficient. Next, it is also shown that for any given g, g≡3 (mod 6) and g≥15, if there exists a GS(2,3,n,g) for all n, n≡1 (mod 2) and g+2≤n≤7g+6, then the necessary conditions are also sufficient. Finally, as an application, it is proved that the necessary conditions for the existence of a GS(2,3,n,g) are also sufficient for g=12,15.展开更多
基金Supported by YNSFC(10001026)for the first authorby Tianyuan Mathematics Foundation of NNSFCGuangxi Science Foundation and Guangxi Education Committee for the second author.
文摘Generalized Steiner triple systems, GS(2, 3, n, g) are equivalent to (g+1)-ary maximum constant weight codes (n, 3,3)s. In this paper, it is proved that the necessary conditions for the existence of a GS(2,3, n, 10), namely, n ≡ 0,1 (mod 3) and n ≥ 12, are also sufficient.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10971051 and 11071056)
文摘A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposable and denoted by IDLSTSx（v） if there does not exist an LSTSx, （v） contained in the collection for any λ 〈 λ. In this paper, we show that for λ = 5, 6, there is an IDLSTSλ（v） for v ≡ 1 or 3 （rood 6） with the exception IDLSTS6（7）.
文摘Generalized Steirier triple systems, GS(2,3,n,g), are equivalent to maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The necessary conditions for the existence of a GS(2,3,n,g) are (n-1)g≡0 (mod 2), n(n-1)g2≡0 (mod 6), and n≥g+2. These necessary conditions are shown to be sufficient by several authors for 2≤g≤11. In this paper, three new results are obtained. First, it is shown that for any given g, g≡0 (mod 6) and g≥12, if there exists a GS(2.3.n.g) for all n, g+2≤n≤7g+13. then the necessary conditions are also sufficient. Next, it is also shown that for any given g, g≡3 (mod 6) and g≥15, if there exists a GS(2,3,n,g) for all n, n≡1 (mod 2) and g+2≤n≤7g+6, then the necessary conditions are also sufficient. Finally, as an application, it is proved that the necessary conditions for the existence of a GS(2,3,n,g) are also sufficient for g=12,15.
