Noboru Hamada

On a geometrical method of construction of maximal t-linearly independent sets

Abstract The problem of constructing a maximal t-linearly independent set in V(r; s) (called a maximal Lt(r, s)-set in this paper) is a very important one (called a packing problem) concerning a fractional factorial design and an error correcting code where V(r; s) is an r-dimensional vector space over a Galois field GF(s) and s is a prime or a prime power. But it is very difficult to solve it and attempts made by several research workers have been successful only in special cases. In this paper, we introduce the concept of a {Σα=1k wα, m; t, s}-min · hyper with weight (w1, w2,…, wk) and using this concept and the structure of a finite projective geometry PG(n − 1, s), we shall give a geometrical method of constructing a maximal Lt(t + r, s)-set with length t + r + n for any integers r, n, and s such that n ⩾ 3, n − 1 ⩽ r0 ⩽ n + s − 2 and r1 ⩾ 1, where r = (r1 + 1)vn−1 − r0 and v n = (s n − 1) (s − 1) .

Construction of Optimal Linear Codes Using Flats and Spreads in a Finite Projective Geometry

In this paper, we shall consider a problem of constructing an optimal linear code whose code length n is minimum among (*, k, d ; s )-codes for given integers k, d and s. In [5], we showed that this problem is equivalent to Problem B of a linear programming which has some geometrical structure and gave a geometrical method of constructing a solution of Problem B using a set of flats in a finite projective geometry and obtained a necessary and sufficient conditions for integers k, d and s that there exists such a geometrical solution of Problem B for given integers k, d and s. But there was no space to give the proof of the main theorem 4.2 in [5]. The purpose of this paper is to give the proof of [5, Theorem 4.2], i.e. to give a systematic method of constructing a solution of Problem B using flats and spreads in a finite projective geometry.

Construction of Optimal Codes and Optimal Fractional Factorial Designs Using Linear Programming

Publisher Summary This chapter describes construction of optimal codes and optimal fractional factorial designs using linear programming. Some theory also discussed. The chapter also discusses the problems and its solution.

Design of a new balanced file organization scheme with the least redundancy

A new balanced file organization scheme of order two for binary-valued records is given which we call HUBFS 2 (Hiroshima University balanced file organization scheme). It can be constructed for a wide range of parameters as is in NBFS 2 . Moreover, it has the least redundancy among the file organization schemes of order two under a general class of probability distribution of records having the invariance property in permutation of attributes. Our scheme is superior to the corresponding NBFS 2 due to Chow as well as BFS 2 due to Abraham, Ghosh, and Ray-Chaudhuri so far as the redundancy is concerned.

Intrarenal Distribution of Blood Flow in Man A New Analytical Method for Dye-Dilution Curves

To measure blood flow through fast and slow pathways of the kidney separately, a new method of analyzing dye-dilution curves was devised. Twenty-five patients with hypertension or renal disease were selected for study. Indocyanine green was injected into one renal artery and dye-dilution curves were recorded in blood from the ipsilateral renal vein, using a densitometer. Recirculation effect was eliminated by the curve obtained from the contralateral renal vein. For analyzing the dyedilution curves, a new mathematical model and an iterative least squares method of fitting the curve using a digital computer were employed. Basic assumption for the model was that the transit time of each molecule of the dye through fast and slow pathways was a random variable and followed a log-normal distribution. The fast flow measured by the new method was significantly decreased in 11 patients with azotemia compared with that in 14 without azotemia, while changes in slow flow were not significant. Consequently, the ratio of the slow flow to the total flow was increased in renal failure.

On the block structure of BIB designs with parameters v = 22, b = 33, r = 12, k = 8, and λ = 4

Abstract It is unknown whether a BIB design with parameters (22, 33, 12, 8, 4) exists or not. In this paper, a necessary condition for the block structure of BIB designs is given. Using this necessary condition, it is shown that the intersection numbers { n i } for any block in any BIB design with parameters (22, 33, 12, 8, 4) must be one of the four types shown in Table I if the BIB design exists. It is also shown that if there exists a BIB design with parameters (22, 33, 12, 8, 4), the BIB design must contain at least one block of Type 1 or 3 in Table I.

Partition of a query set into minimal number of subsets having consecutive retrieval property

Abstract This paper is concerned with information retrieval. The basic problem is how to store large masses of data in such a way that whenever information regarding some particular aspect of the data is needed, such information is easily and efficiently retrieved. Work in this field is thus very important for organizations dealing with large classes of data. The consecutive retrieval (C-R) property defined by S.P. Ghosh is an important relation between a set of queries and a set of records. Its existence enables the design of information retrieval system with a minimal search time and no redundant storage in that the records can be organized in such a way that those pertinent to any query are stored in consecutive storage locations. The C-R property, however, can not exist between every arbitrary query set and every record set. A subset of the query set Q having the C-R property is called a C-R subset and a C-R subset having the maximum cardinality is called the maximal C-R subset. A partition of Q is called the C-R partition if every subset has the C-R property. A C-R partition with minimum number of subsets is called the minimal C-R partition. With respect to the set of all binary queries and the set of all binary records, it is shown that the maximal cardinality of a C-R subset is 2 l -1 where l is the number of attributes concerned. A combinatorial characterization of a maximal C-R subset is also given. A lower bound on the number of subsets in a C-R partition and several examples which attain the lower bound are given. A general procedure for obtaining a minimal C-R partition which attains the lower bound is given provided the number of attributes is even.

The geometric structure and the p-rank of an affine triple system derived from a nonassociative moufang loop with the maximum associative center

Abstract H. P. Young showed that there is a one-to-one correspondence between affine triple systems (or Hall triple systems) and exp. 3-Moufang loops (ML) . Recently, L. Beneteau showed that (i) for any non-associative exp. 3- ML ( E , · ) with ‖ E ‖ = 3 n , 3 ⩽ ‖ Z ( E )‖ ⩽ 3 n −3 , where n ⩾ 4 and Z ( E ) is an associative center of ( E , ·), and (ii) there exists exactly one exp. 3- ML , denoted by ( E n , ·), such that ‖ E n ‖ = 3 n and ‖ Z ( E n )‖ = 3 n −3 for any integer n ⩾ 4. The purpose of this paper is to investigate the geometric structure of the affine triple system derived from the exp. 3- ML ( E n , ·) in detail and to compare with the structure of an affine geometry AG ( n , 3). We shall obtain (a) a necessary and sufficient condition for three lines L 1 , L 2 and L 3 in ( E n , ·) that the transitivity of the parallelism holds for given three lines L 1 , L 2 and L 3 in ( E n , ·) such that L 1 ‖ L 2 and L 2 ‖ L 3 and (b) a necessary and sufficient condition for m + 1 points in E n (1 ⩽ m n ) so that the subsystem generated by those m + 1 points consists of 3 m points. Using the structure of hyperplanes in ( E n , ·), the p -rank of the incidence matrix of the affine triple system derived from the exp. 3- ML ( E n , ·) is given.