golay codes

Question 1

binary golay code:

Answer 1

denoted G23, cyclic code in (F2)^23 generated by g(x), or any code linearly equivalent to it (such as by writing the codevectors backwards), a perfect [23, 12 7]2 code

Question 2

extended code:

Answer 2

if C is a binary code of length n, we define the extended code Ĉ of length n+1 as {ĉ: c in C}

Question 3

extended vector:

Answer 3

a vector y=(y1,y2,…,yn) is extended to ŷ=(y1,…,y(n+1)) where y(n+1)=y1+…+yn - essentially, append one bit so the resulting vector has even weight, so ŷ=(y,0) if y is even and (y,1) if y is odd

Question 4

overlap:

Answer 4

if u,v in F2, the overlap is the number of positions i such that ui=vi=1

Question 5

properties of the overlap:

Answer 5

w(u+v)=w(u)+w(v)-2overlap(u,v)
u.v <=> overlap(u,v) is even
w(u), w(v) are multiples of 4 and u.v=0 => w(u+v) is a multiple of 4

Question 6

extended binary golay code:

Answer 6

denoted g24, not cyclic, but to find a generator matrix write g23 and append a column of 1s or 0s (whatever makes the rows even) to the right, is self-dual (G24=(G24)⊥, the weight of every codevector is a multiple of 4

Question 7

ternary golay code:

Answer 7

denoted G11, the cyclic code in (F3)^11 generated by g(x) or any linearly equivalent code, a perfect [11, 6, 5]3 code

Question 8

parameter equivalent:

Answer 8

two codes are parameter equivalent if they are both [n, k, d]q codes for some n, k, d, and q

Question 9

classification of perfect codes where q is a prime power:

Answer 9

let q be a power of a prime number. a perfect [n, k, d]q code is parameter equivalent to one of the following:
a trivial code (n arbitrary, k=n, d=1, q=any prime power)
a binary repetition code of odd length (n=odd, k=1, d=n, q=2)
a hamming code (n=(q^(r)-1)/q-1, k=n-r, d=3, q=any prime power)
the golay code G23 ([23, 12, 7]2)
the golay code G11 ([11, 6, 5]3)