A New Construction for the Extended Binary Golay Code
نویسندگان
چکیده
We give a new construction of the extended binary Golay code. The construction is carried out by taking the Gray image of a self-dual linear code over the ring R = F2+uF2+vF2+uvF2 of length 6 and size 212. Writing a typical generating matrix of the form [I3|A], with A being a 3× 3 matrix over R, and finding some dependencies among the entries of A, we are able to set a general form for the generating matrices of self-dual codes of length 6. Using some special properties of elements of R, we end up with a family of generating matrices all of which give us the extended binary Golay code. We also prove the minimum distance property analytically.
منابع مشابه
A generalisation of Turyn’s construction of self-dual codes
In [17] Turyn constructed the famous binary Golay code of length 24 from the extended Hamming code of length 8 (see also [10, Theorem 18.7.12]). The present note interprets this construction as a sum of tensor products of codes and uses it to construct certain new extremal (or at least very good) self-dual codes (for example an extremal doubly-even binary code of length 80). The lattice counter...
متن کاملConstructing the Extended Binary Golay Code
Coding theory is the subject which is concerned with how information can be sent over a noisy channel. A code then is a collection of codewords which are strings of a fixed number of letters from an alphabet. Some of these strings are codewords others are not. When a codeword is sent over a channel there is a probability less than /2 that each letter in the codeword will be changed, thus introd...
متن کاملThe Golay Code Outperforms the Extended Golay Code Under Hard-Decision Decoding
We show that the binary Golay code is slightly more power efficient than the extended binary Golay code under maximum-likelihood (ML), hard-decision decoding. In fact, if a codeword from the extended code is transmitted, one cannot achieve a higher probability of correct decoding than by simply ignoring the 24th symbol and using an ML decoder for the non-extended code on the first 23 symbols. T...
متن کاملThe poset structures admitting the extended binary Golay code to be a perfect code
Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57–72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that th...
متن کاملImplementation of Effective Error Correction Architecture for (24,12) Extended Golay Code
With the increase in the technological advancements we are now capable of designing and using high speed communication systems and memory systems with very huge capacities. All these systems work on binary data and there is a great need to protect this information from being corrupted for which we are now using Error Correcting Codes (ECCs). The (24,12) Extended Golay Code is one of the widely ...
متن کامل