Free download скачать Algebraic Coding Theory and Applications by G. Longo
English | PDF | 1979 | 534 Pages | ISBN : 3662387522 | 28.2 MB
The  last twenty-fit,e years have witnessed  thr:  growth  of  one  of  the  most  elegant  and esoteric branches  of  applied  mathematics:  Algebraic Coding Theory.  Areas  of  mathematics which were previously considered to be  of  the  utmost  purity  have been applied  to  the problem  of  constructing  error-correcting codes and their decoding algorithms. In spite  of  the impressive theoretical  accomplishments  of  these  twenty-five  years, however,  only  recently has algebraic coding  been  put  into  practice.

To  present  some  of  the latest results  on  the  theory  and applications  of  algebraic coding, a  number  of  scholars  who  have  been  active  in  the  various areas  of  coding research were invited to lecture at the  summer  school  on  "Algebraic Coding:  Theory  and Applications",  organized  by  Giuseppe  Longo  at the Centre International des Sciences Mecaniques  (ClSM)  in Udine, a  picturesque  city  in  northern Italy,  for  a  period  of  two  weeks ill  July,  1978.
The  first  contribution,  "A  Survey  of  Error-Control  Codes",  by  P.G. Farrell  (the University  of  Kent, Great Britain)  is  an  excellent  compilati01l  and  condef'sation  of numerous  results  on  error-correcting codes. This  contribution  consists  of  four  main sections. The  first introduces the reader to the basic facts  about  error-correcting codes, the  second .describes various decoding  methods,  the third lists  some  classes  of  error-control codes which hat'e  foulld  practical application, and the last  is  devoted  to the  performance  of  such codes.
The  second  contributioll,  "The  Bounds  of  Delsarte  and  LovGsz, and  Their  Applications to Coding  Theory",  is  by  R.J.  McEliece (University  of  Illinois, U.S.A.). In  1972,  P.  Delsarte developed  a  new  powerful  technique  for  obtaining  upper  bounds  on  the largest possible number  of  codewords  in  a  code  of  fixed  length and  minimum  Hamming  distance. This technique  is  nowadays  usually called the linear programming approach. In  1977,  L.  Lovasz produced  all  astonishingly simp! '  50114  ti01l  to a long-standing  problem  in  information  theory which was  posed  by  C.  ShamlO1I  ill  1956,  namely  the  problem  of  computing  the zero-error capacity  of  a certaill discrete  memoryless  channel having five  inputs  and  outputs.  Lovasz's technique  call be applied to any graph (or discrete  memoryless  channel), although in general it gilles  only  all  upper  bound  for  the "Shallnon  capacity",  rather than the true value. In his paper, McElicce  offers  a  unified  treatment  of  these  two  techniques using standard  methods of  linear  a l ~ e b r a .  The result  is  an  extremely powerful and general technique for studying combinatorial packing problems. This technique  is  used  to  obtain,  as  special  cases,  the McEliece-Rodemich-Rumsey-Welch bound for binary codes and Lovasz's bound on the "Shannon capacity"  of  many graphs.
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