New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra

We use the Terwilliger algebra to provide a new approach to the Assmus-Mattson theorem. This approach also includes another proof of the minimum distance bound shown by W.J. Martin as well as its dual.

[1]  N. J. A. Sloane,et al.  A strengthening of the Assmus-Mattson theorem , 1991, IEEE Trans. Inf. Theory.

[2]  John S. Caughman Spectra of Bipartite P- and Q-Polynomial Association Schemes , 1998 .

[3]  J. Simonis MacWilliams identities and coordinate partitions , 1995 .

[4]  Paul M. Terwilliger The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph , 2005, Graphs Comb..

[5]  Damir Čemerin,et al.  IV , 2011 .

[6]  Tatsuro Ito,et al.  Some algebra related to P- and Q-polynomial association schemes , 1999, Codes and Association Schemes.

[7]  William J. Martin,et al.  Minimum Distance Bounds for s-Regular Codes , 2000, Des. Codes Cryptogr..

[8]  Kenichiro Tanabe A New Proof of the Assmus–Mattson Theorem for Non-Binary Codes , 2001, Des. Codes Cryptogr..

[9]  IV John S. Caughman The Terwilliger algebras of bipartite P - and Q -polynomial schemes , 1999 .

[10]  Junie T. Go The Terwilliger Algebra of the Hypercube , 2002, Eur. J. Comb..

[11]  H. Mattson,et al.  New 5-designs , 1969 .

[12]  Arlene A. Pascasio On the Multiplicities of the Primitive Idempotents of a Q-Polynomial Distance-regular Graph , 2002, Eur. J. Comb..

[13]  Paul M. Terwilliger The Subconstituent Algebra of an Association Scheme, (Part I) , 1992 .

[14]  Alexander Schrijver,et al.  New code upper bounds from the Terwilliger algebra and semidefinite programming , 2005, IEEE Transactions on Information Theory.

[15]  Shirley Dex,et al.  JR 旅客販売総合システム(マルス)における運用及び管理について , 1991 .

[16]  IV JohnS.Caughman The Terwilliger algebras of bipartite P- and Q-polynomial schemes , 1999, Discret. Math..

[17]  Philippe Delsarte,et al.  On Error-Correcting Codes and Invariant Linear Forms , 1993, SIAM J. Discret. Math..

[18]  P. Delsarte AN ALGEBRAIC APPROACH TO THE ASSOCIATION SCHEMES OF CODING THEORY , 2011 .

[19]  Alexander Schrijver,et al.  New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming , 2006, J. Comb. Theory, Ser. A.

[20]  Remarks on tactical configurations in codes , 1980 .

[21]  Paul M. Terwilliger,et al.  The Subconstituent Algebra of an Association Scheme (Part III) , 1993 .

[22]  Marc Lalaude-Labayle On binary linear codes supporting t-designs , 2001, IEEE Trans. Inf. Theory.

[23]  William J. Martin Mixed block designs , 1998 .

[24]  William J. Martin,et al.  Designs in Product Association Schemes , 1999, Des. Codes Cryptogr..

[25]  Philippe Delsarte,et al.  Association Schemes and t-Designs in Regular Semilattices , 1976, J. Comb. Theory A.

[26]  E. Bannai,et al.  Algebraic Combinatorics I: Association Schemes , 1984 .

[27]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[28]  KENICHIRO TANABE A Criterion for Designs in ℤ4-codes on the Symmetrized Weight Enumerator , 2003, Des. Codes Cryptogr..

[29]  Christine Bachoc On Harmonic Weight Enumerators of Binary Codes , 1999, Des. Codes Cryptogr..

[30]  Kevin T. Phelps,et al.  Kernels and p-Kernels of pr-ary 1-Perfect Codes , 2005, Des. Codes Cryptogr..

[31]  Henry Beker 2-Designs Having an Intersection Number k - n , 1980, J. Comb. Theory, Ser. A.