Group invariant weighing matrices

We investigate the existence problem of group invariant matrices using algebraic approaches. We extend the usual concept of multipliers to group rings with cyclotomic integers as coefficients. This concept is combined with the field descent method and rational idempotents to develop new non-existence results.

[1]  K. T. Arasu,et al.  Perfect Ternary Arrays , 1999 .

[2]  K. T. Arasu,et al.  Some New Results on Circulant Weighing Matrices , 2001 .

[3]  Bernhard Schmidt,et al.  The Field Descent Method , 2005, Des. Codes Cryptogr..

[4]  T. Y. Lam,et al.  On Vanishing Sums of Roots of Unity , 1995 .

[5]  V. Yorgov,et al.  On the Existance of Certain Circulant Weighing Matrices , 2012 .

[6]  R. Turyn.  Character sums and difference sets. , 1965 .

[7]  K. Arasu,et al.  Nonexistence of CW(154,36) and CW(170,64) , 2011 .

[8]  Hanfried Lenz,et al.  Design theory , 1985 .

[9]  Ming Ming Tan.  Relative difference sets and circulant weighing matrices , 2014 .

[11]  K. Arasu,et al.  Nonexistence of two circulant weighing matrices of weight 81$^{1}$ , 2015 .

[12]  Jennifer Seberry,et al.  Weighing matrices and their applications , 1997 .

[13]  Robert Craigen,et al.  The structure of weighing matrices having large weights , 1995, Des. Codes Cryptogr..

[14]  Jennifer Seberry,et al.  Orthogonal designs III: Weighing matrices , 1974 .

[15]  J. Cassels.  On a conjecture of R.M. Robinson about sums of roots of unity. , 1969 .

[16]  Erwin Schrödinger International,et al.  Supported by the Austrian Federal Ministry of Education, Science and Culture , 1689 .

[17]  Hiroyuki Ohmori.  Classification of weighing matrices of order 13 and weight 9 , 1993, Discret. Math..

[18]  Orthogonal designs H , 1975 .

[19]  K. T. Arasu,et al.  Nonexistence of CW(110, 100) , 2012, Des. Codes Cryptogr..

[20]  Bernhard Schmidt,et al.  Circulant weighing matrices whose order and weight are products of powers of 2 and 3 , 2013, J. Comb. Theory, Ser. A.

[21]  Ali Nabavi,et al.  Determination of all possible orders of weight 16 circulant weighing matrices , 2006, Finite Fields Their Appl..

[22]  Richard Hain.  Circulant weighing matrices , 1977 .

[23]  Bernhard Schmidt,et al.  Cyclotomic integers and finite geometry , 1999 .

[24]  R. Mullin,et al.  A note on balanced weighing matrices , 1975 .

[25]  J. Seberry,et al.  Hadamard matrices, Sequences, and Block Designs , 1992 .

[26]  K. T. Arasu,et al.  Study of proper circulant weighing matrices with weight 9 , 2008, Discret. Math..

[27]  R. McFarland,et al.  On multipliers of abelian difference sets , 1970 .

[28]  Michel Kervaire,et al.  A new restriction on the lengths of golay complementary sequences , 1990, J. Comb. Theory A.

[29]  R. Mullin,et al.  On the Nonexistence of a Class of Circulant Balanced Weighing Matrices , 1976 .

[30]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[31]  Louis Pigno.  A multiplier theorem. , 1970 .

[32]  Robert Craigen,et al.  Hadamard Matrices from Weighing Matrices via Signed Groups , 1997, Des. Codes Cryptogr..

[34]  Ali Nabavi,et al.  Circulant weighing matrices of weight 22t , 2006, Des. Codes Cryptogr..

[35]  Jennifer Seberry,et al.  Circulant weighing designs , 1996 .

[36]  K. T. Arasu,et al.  Group developed weighing matrices , 2013, Australas. J Comb..

[37]  Bernhard Schmidt,et al.  Characters and Cyclotomic Fields in Finite Geometry , 2002 .

[38]  Jennifer Seberry,et al.  On the weighing matrices of order 4n and weight 4n-2 and 2n-1 , 1995, Australas. J Comb..

[39]  L. Epstein,et al.  The Classification of Circulant Weighing Matrices of Weight 16 and Odd Order , 1999 .

[40]  Albert Leon Whiteman,et al.  Some results on weighing matrices , 1975, Bulletin of the Australian Mathematical Society.

[41]  Jennifer Seberry,et al.  Circulant weighing matrices , 2010, Cryptography and Communications.

[42]  Siu Lun Ma,et al.  Polynomial addition sets , 1985 .

[43]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[44]  Siu Lun Ma,et al.  Proper circulant weighing matrices of weight $$p^2$$ , 2014, Des. Codes Cryptogr..