Justesen codes are created by tilting (n,k) Reed Solomon codes over GF(2^m) into (mn,mk) binary codes for multiple burst error correction. They are very good for long block lengths. Alternant codes are variation of BCH codes with fixed rate and large minimum distance.They are subfield-subcode of a Reed-Solomon codes over GF(q^m).Goppa codes are designed distance d have additional property over Alternant codes that inverse frequency template has width d...

1. Justessan Codes
Alternant Codes
Goppa codes

2. Justesen Codes
 Mapping codes converts (N,K) linear code over GF(qm
) into
(mN,mK) linear code over GF(q) by ‘tilting’ each qm
-ary
symbol into m q-ary symbols.
 d* nonzero symbols of min wt. codeword tilts to md*
symbols, not all symbols zero.
 Code rate remains unchanged but minimum distance is
much smaller fraction of block length.
 Justesen modified the construction to give a good code for
long block lengths.
 Advantage: Creates a simple form of multiple burst-error
correcting codes.
 Gives infinite family of good binary codes.

3. Justesen Codes - Construction
 Construction starts with all the codewords of a single fixed
(N,K) Reed-Solomon code that has the same number of
codewords as desired Justesen code.
 Let α be the primitive element of GF(qm
).
 Starting with codeword C = (c0, c1, … cN-1) from Reed-
Solomon code, first form the 2 by N array of GF(qm
)-ary
symbols. (ci Є GF(qm
) )
 c0 c1 c2 … cN-1
 α0c0 α1c1 α2c2 … αN-1cN-1
 C’ = (c0,α0c0 ; c1,α1c1 ; c2,α2c2 ; … cN-1,αN-1cN-1)
 Replace each element by binary m-tuple symbols we obtain
binary vector of length 2mN.
 This gives one code word of the Jestesen Code.
 Code rate R = K/2N, half of RS code.

4. Justesen Codes
 Justesen code is the concatenation code of an RS code and
linear binary codes.
 A natural decoding algorithm for Justesen code would be
one that reverses the process of concatenation.
 Algorithm:
 Decode each set of m bits to yi over GF(qm
).
 Ties are broken arbitrarily to give two set.
 Decode y = (y0, . . . , yn−1) using any decoding algorithms for
RS code.
 The decoding algorithm can be used to correct all errors up
to less than dD/4 in number.

5. Justesen Codes
 Theorem: The minimum distance of the (2mN, mK)
Justesen code constructed from an (N,K) RS code is
bounded by --

6. Justesen Codes
 Proof:
 The minimum weight RS codeword has N-K+1 distinct non-
zero symbols.
 These will make N-K+1 nonzero pairs (ci,αici ) which appear
as distinct 2m-tuples.
 The weight is atleast as large as word constructed by filling
N-K+1 2m-tuples by N-K+1 distinct 2m-tuples of smallest
weight.
 In an m-tuple, there are (2
i
m) ways of picking i non-zero
places and (q-1) different non-zero values at each place.
 Hence there is a weight I for every I that satisfies (2).
 The minimum distance is at-least as large as the sum of the
weights of these pairs. Hence (1).

7. Justesen Codes
 Disadvantage: Not very attractive for random error
correction.
 Good only for long block-lengths.
 Hence did not get popular.
 Hence they do not have well developed collection of
decoding algorithms.

8. Alternant Codes- Limitations of BCH codes:
 A BCH code over GF(q) of block length n = qm-1 is a
subfield-subcode of a Reed-Solomon code over GF(qm).
 subfield-subcode has same length as original but fewer
codes.
 BCH code consists of all those Reed-Solomon codes that
are GF(q)-valued.
 BCH codes with large block length and large minimum
distance contain fewer codes.
 But in BCH code, with increasing block length and fixed rate
R’ (k/n≥R’),normalized minimum distance d*/n approaches
zero.
 Hence, the BCH code uses very few of many code words of
RS code or else has poor distance structure.
 Alternant codes, a variation of BCH code, increases
minimum distance by reducing RS code to a subfield by
new way.

9. Alternant Codes
 Alternant codes are linear codes that are a variation of BCH
codes defined such that in spite of fixed rate, large minimum
distance can be obtained.
 Let n = qm-1,
 A fixed n-vector h of nonzero components over GF(qm)
called (time-domain) template, is chosen.
 Reed-Solomon code over GF(qm) with designed distance
2t+1 is chosen.
 If Alternant code is GF(q)-valued vectors C and C’ is
codeword in RS code then-
 c’i = ci hi for i= 0,1,…n-1 . (component-wise in Time domain)
 hi is always nonzero.
 So, let gi = hi
-1.
 ci = gic’I for i= 0,1,…n-1 .

10. Alternant Codes
 Minimum distance very large if template chosen properly.
 Can be defined in frequency domain also.
 Let h ≠ 0 and H is its frequency domain template.
 Cyclic convolution H*C gives Reed-Solomon spectrum.
∑n-1
k=0
H((j-k)) Ck = 0 j = j0, … j0+ 2t-1.
 If G is transform of vector gi ( =hi
-1), H*G is a delta function.
 (If j=0, (H*G)j =1; otherwise (H*G)j =0.)
 As polynomial this convolution becomes
 H(x)G(x) = 1 (mod xn-1).
 H(x) is a polynomial over GF(q), it has no roots in GF(qm).
 Hence H(x) is prime to xn-1.
 H(x)G(x) + (xn-1) F(x) =1
 Hence H(x)G(x) = 1 (mod xn-1).

11. Alternant Codes
 The alternant code C(α, h ) consists of all codewords of
GRSk0
which have components from GF(q).
 C(α,h) consists of all vectors C over GF(q) such that CHT=0,
where H is given by H = Vr(α1 …αn) .diag(h1 …hn)
1 … 1 h1 0 . 0
α1 … αn 0 h2 . 0
H= α1
2 . αn
2 . 0 . 0
. . . . . . .
(α1)r-1 … (αn)r-1 0 0 . hn
h1 … hn
h1α1 … hnαn
H= . . .
. . .
h1(α1)r-1 … hn(αn)r-1

12. Alternant Codes - example
 Decoding by any method used for BCH or RS codes.
GF(8)
BCH Code Reed-Solomon Code (7, 5, 3) Alternant code
g=(5, 6, 1, 4, 1, 1, 7)
0000000 0000000 0000000
0000163
0000276
0000315
0001011 0001011
0001172
0001304
0007077 0001011
0007114 0001111
0007217
0010073
0010110 0010110 0010110

13. Goppa Codes
 Goppa code is the generalization of the class of BCH code.
 BCH code can also be described alternately as –
 Let a(x) and G(x) be two polynomials over same field with
no common factors.
 By Euclid”s algorithm, a polynomial u(x) exists such that –
 a(x) u(x) Ξ 1 mod G(x)
 One option u(x) can be a-1(x) -- can be found under
modG(x).
 THEOREM:- The q-ary narrow sense BCH code of length n
with designed distance d is equivalent to code
 { c єVn(q) ‫׀‬∑i=0
n-1
ci / (x – αi’ ) Ξ 0 (mod xd -1)}
 -where α є GF(qm).
 PROOF--

14. Goppa Codes
 PROOF—
 C will be a code of BCH if S1+ S2+ S3… S2t= 0
 ∑j=1
d-1
Sj= 0 (2t = d-1)

 Hence S(x) =0 where S(x) = ∑j=1
d-1
Sjxj-1
with Sj= c(αj )
 S(x) Ξ ∑j=1
d-1
∑i=0
n-1
ci αij xj-1
αi +α2i +α3i +…α(d-1)i
 Ξ ∑i=0
n-1
ci ∑j=1
d-1
αij xj-1
Take αi out and expand the rest.
 Ξ ∑i=0
n-1
ci αi[1+αi x+α2i x2
+… +α(d-1)i xd-1
] GP
 Ξ ∑i=0
n-1
ci αi[1- α(d-1)i xd-1
]/ [1-αi x] mod(xd-1
)

15. Goppa Codes
 Ξ ∑i=0
n-1
ci αi [1- α(d-1)i xd-1
]/ [1-αi x] mod(xd-1
)
 Ξ ∑i=0
n-1
ci αi [1]/ [1-αi x] mod(xd-1
) 2nd term remainder is 0.
 Ξ ∑i=0
n-1
ci / [α-i - x] mod(xd-1
)
 Ξ - ∑i=0
n-1
ci / [x-α-i] mod(xd-1
)
 Replacing α by 1/α will not alter the summation.
 Hence proved for BCH code.

16. Goppa Codes
 Goppa code of designed distance d is an alternant code of
designed distance d, with additional property that—
 The inverse frequency template G has width d.
 Inverse frequency template, called Goppa polynomial can
be described as G(x) with degree d-1.
 A narrow sense Goppa code is a Goppa code with 2t parity
frequencies at locations α0, …..αn-2t+2, αn-2t+1
 THEOREM:- Let L = {α0 , α1 , …, αn-1} be a subset of GF(qm)
of size n,and let G(x) be a polynomial of degree s over
GF(qm) but has no root from L then—Goppa code is given
by
 Ғ(L,G) = { c єVn(q) ‫׀‬∑i=0
n-1
ci / (x – αi ) Ξ 0 (mod G(x))}
 Taking G(x) = xd-1
–
 Goppa codes contain BCH code as subclass.

17. Goppa Codes
 THEOREM:- The narrow sense Goppa code over GF(q)
with block length n = qm -1 and with Goppa polynomial G(x)
is given by the set of all vectors c = (c0, …cn-1) over GF(q)
satisfying
 ∑i=0
n-1
ci’ Π (x – α-i) = 0 (mod G(x)) excluding self i≠i’

18. Goppa Codes -Example
 Smallest example is (8, 2, 5) binary Goppa code.
 G(x) = x2 + x + 1.
 Roots(zeros) of G(x) are in GF(4) or in any extension of
GF(4).
 Hence none are in GF(8).
 Hence G(x) can be used to obtain a Goppa code with
blocklength 8, minimum distance of 5 and 2 information
symbols.

19. Goppa Codes -Example
 Example: Let α be the primitive element in GF(24) satisfying
α4 + α3 + 1=0. Consider the binary Goppa code Ғ(L,G) of
length 12 with G(x)=(x+ α)(x+ α14) and L={αi ‫׀‬ 2 ≤ i ≤ 13}.
Find parity check matrix H.
 G(x)=(x+ α)(x+ α14)= x2 + α8x + 1
 α and α14 are used to generate G(x), while rest of 12
elements are member of L.
 ∑i=0
n-1
ci / (x – αi ) Ξ 0 (mod G(x))} for ci to be code.
 Product of C and H should be zero for correct code ,
otherwise would give syndrome.
 Hence H becomes inverse of (x – αi ). {αi ‫׀‬ 2 ≤ i ≤ 13}.
 e.g. Lets find inverse of (x – α2 ) modulo G(x).
 Let inverse of (x – α2 ) modulo G(x) is ax+b.
 (x – α2 )(ax+b) Ξ 1(mod G(x))

20. Goppa Codes -Example
 (x – α2 )(ax+b) = ax2 + (b + aα2)x + bα2
 G(x)=x2 + α8x + 1=0
 x2 = α8x + 1
 = a (α8x + 1) + (b + aα2)x + bα2
 = (aα10 + b)x + a + bα2 Ξ 1 (mod G(x))
 Hence ---- (aα10 + b) =0 and a + bα2 = 1
 Solving above a= α14 and b= α9
 Inverse ax+b = α14 x+ α9
 Similarly 11 other inverses can be found.
H= α9 α α8 α13 α7 α5 0 α9 α α6 α5 α6
α14 α3 α α4 α7 α1 α14 α4 α14 α9 α9 α3

21. Goppa Codes -Example
1 0 0 0 1 1 0 1 0 1 1 1
0 1 1 1 1 1 0 0 1 1 1 1
1 0 1 1 1 0 0 1 0 1 0 1
H= 0 0 1 0 0 1 0 0 0 1 1 1
0 0 0 1 1 0 0 1 0 1 1 0
0 0 1 0 1 1 0 0 0 0 0 0
1 0 0 0 1 0 1 0 1 1 1 0
1 1 0 1 0 0 1 1 1 0 0 1
H= α9 α α8 α13 α7 α5 0 α9 α α6 α5 α6
α14 α3 α α4 α7 α1 α14 α4 α14 α9 α9 α3
H has rank 8 and Goppa code of { , , ≥ )

22. Goppa Codes-- Decoding
 THEOREM:- Let σ(x) and ω(x) be the error locator and error
evaluator polynomials of an error pattern of weight at most
[s/2] where s is degree of polynomial G(x) of Goppa code. (
s= n-k =d+1. i.e. s/2=t)
 Then σ(x) = λ vi(x) and ω(x) = λ si(x)
 where si(x) and vi(x) are obtained from Euclid’s Algorithm
applied to G(x) and S(x) (to find hcf or gcd λ) until
deg(si(x))<[s/2] for the first time and where λ is chosen such
that λ vi(x) is monic.
 (Proof omitted. Ref:
hyperelliptic.org/tanja/teaching/CCI11/CODING.pdf.)
 Roots of σ(x) and ω(x) together will give error location and
magnitude.