Nordstrom-Robinson code and A7-geometry
نویسنده
چکیده
The Nordstrom-Robinson code NR is a non-linear binary code of length 16, with 2 codewords and minimum distance 6. Its automorphism group is a semidirect product of an elementary abelian group of order 16 and the alternating group A7. This group and the corresponding action of A7 is also at the origin of the sporadic A7-geometry. We construct this geometry and derive the Nordstrom-Robinson code from it.
منابع مشابه
On the extention of propelinear structures of Nordstrom-Robinson code to Hamming code
A code is called propelinear if its automorphism group contains a subgroup that acts regularly on its codewords, which is called a propelinear structure on the code. In the paper a classification of the propelinear structures on the Nordstrom-Robinson code is obtained and the question of extension of these structures to propelinear structures of the Hamming code, that contains the Nordstrom-Rob...
متن کاملConstruction of a (64, 237, 12) Code via Galois Rings
Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4m ≥ 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 237, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers...
متن کاملThe Nordstrom-Robinson Code is the Binary Image of 19 the Octacode
The Nordstrom-Robinson code, a nonlinear binary code of length 16 and minimal Hamming distance 6, is the binary image of the octacode, a linear self-dual code over 4 of length 8 and minimal Lee distance 6. Since the octacode is the 4-analogue of a Hamming code, this provides an extremely simple definition of the Nordstrom-Robinson code. A different version of this paper appeared in: Coding and ...
متن کاملA linear construction for certain Kerdock and Preparata codes
The Nordstrom-Robinson, Kerdock and (slightly modified) Preparata codes are shown to be linear over 4, the integers mod 4. The Kerdock and Preparata codes are duals over 4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over 4. This provides a simple definition for these codes, and explains why their Hamming weight distributions are dual to each oth...
متن کاملModular and p-adic Cyclic Codes
This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo pa and over the p-adic numbers, where p is a prime not dividing n. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomial X 3 + ,~X 2 + (L I)X -l, where )~ satisfies ~2 _ k + 2 = 0. This is the 2-adic generalization of both the bina...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Finite Fields and Their Applications
دوره 13 شماره
صفحات -
تاریخ انتشار 2007