Relation between Two Classes of Binary Quasi-Cyclic Goppa Codes

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Relation between Two Classes of Binary
Quasi-Cyclic Goppa Codes Sergey Bezzateev and
Natalia Shekhunova Saint Petersburg State
University of Airspace Instrumentation Saint-Peter
sburg, Russia e-mail bsv_at_aanet.ru, sna_at_delfa.net
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Binary Goppa codes

• A binary vector a(a1,an) of length n is a
  codeword of the ?(L,G) Goppa code if and only if

G(x)- Goppa polynomial, L- locator set.
Parameters of binary Goppa codes Code length n
2m Code dimension kn-mdegG(x) Minimal
distance ddegG(x)1dds.
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Separable Binary Goppa codes

• G(x) is separable polynomial
• Minimal distance for separable binary Goppa codes
  
• d 2degG(x)1dds

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Main questions

• To find codes with
• Improved estimations of parameters dimension and
  minimal distance
• Good parameters (VG bound)
• Special structure (large permutation group,
  quasi-cyclic, etc.)
• ddds
• The binary separable codes with the location set
  L from GF(22l) are of the greatest interest

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Background

• M. Loeloeian and J. Conan (1984) presented
  (55,16,19) Goppa code
• G(x) (x -a9)(x - a12)(x - a30)(x - a34)(x -
  a42)(x - a43)(x - a50)(x - a54)
• where a is a primitive element of GF(26).
  dds17ltd19.
• We (1986) considered, G(x) xt1 Vtxt Vx
  1, Subfield subcodes,
• where L ? GF(t2), V ? GF(t2)\1 , n t2 - t -
  1.
• k t2 - t - 1 - 2l(t - 3/2 ), where t2l . ,
  dds2t3.
• M. Loeloeian and J. Conan (1987), G(x) xt x,
• where L ? GF(t2) and n t2 - t. k t2 - t -
  2l(t - 3/2 ) - 1, dds2t1.
• A.M.Roseiro, J.I.Hall, J.E.Adney and M.Siegel
  (1992)
• By representation Gt(x) G(x) mod (xt2 x), k
  t2 - t - 2l(t - 3/2) - 1, dds2t1.
• We described(1995) G(x) xt-1 1, k t2 - t -
  2l(t - 3/2 ), and we have proved that d2t-1.

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Background

• P. Veron (2001) G(x) xt x - Quadratic trace
  Goppa codes.
• k t2 - t - 2l(t - 3/2 ) - 1, d2t4
• P. Veron (2005) G(x) xt1VtxtVx1.
• k t2 - t - 1 - 2l(t - 3/2)
• and G(x) xt-1 1 , k t2 - t 1- 2l(t - 3/2) -1
• We (1995), P. Veron(1998), G. Bommer and F.
  Blanchet (2000) - all the forementioned codes are
  quasi-cyclic binary Goppa codes.

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• Two classes of quasi-cyclic Goppa codes ?(LG(x))
  and ?(LG(x)), where
• G(x) xt-1 1
• nt2-t1, k t2 - t - 2l(t -3/2), d dds2t - 1
  
• G(x) xt1 1
• nt2-t-1, k t2 - t - 2l(t -3/2)-1, d dds2t
  3

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CODEWORD STRUCTURE OF THE BINARY ?(LG(x)) CODE

• It is easy to show that ?(LG(x)) code is the
  quasi-cyclic code with the length of cycloid (t
  -1) and number of cycloids t.
• The codewords of this code have one fixed
  position 0.
• The total length of the code is
• n t(t - 1) 1

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• The numerators of the codewords of the ?(LG(x))
  code can be represented in the following form
• L ßi, ßi at1 ,, ßi a(t1)(t-2) i1,.. t
  U0
• where ß at-1 ,
• a is the primitive element of GF(22l),
• and ßi, ßi at1 ,, ßi a(t1)(t-2) are
  numerators of positions that form the
  correspondent cycloids.

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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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• ?1(L1G(x)) code obtained as trancated ?(LG(x))
  code by information position 0, i.e., we remove
  all codewords with 1 on position 0 from
  ?(LG(x)) code.
• Then L1 L\0 and ?1(L1G(x)) code is still
  quasi-cyclic code

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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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• It is possible to transform(Lemma 1) parity check
  matrix of the code ?1(L1G(x)) to the a parity
  check matrix of the code ?2(L2G2(x)) where G2(x)
  xt x , L2 L1.
• This code still quasi-cyclic with the length of
  cycloid t -1 and the number of cycloids is t,
• n2 t(t - 1).

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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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• Let us consider now the following substitution x
  -gt z ? , and ? t ? 1 0, ??GF(t2).
• Then G2(x) xt x zt ? t z ? zt z
  1 G3(z).

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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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• The code ?3(L3G3(x)) has parameters
• n3 t(t - 1)
• k3 k2 k1 1
• d3 2t 4

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• Let us consider now ?3(L3 G3(x)) code obtained
  from ?3(L3 G3(x))-code by trancation on position
  0, i.e., L3 L3\0
• This code has parameters
• n3 n3 - 1
• k3 k3 k2 k1 - 1
• d3 d3 - 1 d2 - 1 2t 3

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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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• Now let us use the following substitution
• z -gt y -1 . Then
• G3(z) zt z 1 y-t y-1 1 -gt
• G4(y) yt yt-1 1.
• Code ?4(L4G4(x)) has parameters
• n4 n3 n3 - 1
• k4 k3
• d4 d3 - 1

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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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• Lemma 4
• Code ?4 (L4 G4(x)) ?5(L5 G5(x)) where
  G5(y) yG4(y) yt1yty and L5 L4 .

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Transformation from ?(LG(x)) to ?(LG(x))
G(x) xt-1 1
truncate
G1(x) G (x), L1 L\0
Lemma1
G2(x) xG1(x), L2 L1
Theorem 2 (d22t4)
Substitution x-gt x? , ?t ? 1 0, ??GF(t2)
G3(x) xt x1, L3
truncate
G3(x) xt x1, L3 L3 \0
Lemma 3
Substitution x -gt x-1
G4(x) xt xt-1 1, L4
Lemma 4
G5(x) xG4(x), L5 L4
Substitution x -gt x1
G6(x) xt1 1, L6
Theorem 1 (d22t3)
G(x) xt1 1
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• Let us use the following substitution
• y -gt u 1 then
• G5(y) yt1 yt y -gt (u 1)t1 (u 1)t
  u ut1 1 G6(u)G(u).

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• ?(LG(x)) codes have the following parameters
• n n4 n3 -1 t(t - 1) - 1
• k k4 k3 k - 1
• Theorem 1
• The minimal distance of ?(LG(x)) code is
• d 2t 3
• Theorem 2
• The minimal distance of ?2(L2G2(x)) code is
• d2 2t 4 and number of information symbols is
  k2 k1 - 1.
• d2 d3 d3 1 d 12t 4

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Codes parameters
code code length code dimension minimal distance
G (x) xt-1 1 t2 t1 t2 t-2l(t-3/2) (P. Veron, 2005) 2t-1
G2(x)xt x t2 t t2 t-1-2l(t-3/2) (P.Veron,2001) 2t4
G (x) xt1 1 t2 t-1 t2 t-1-2l(t-3/2) (P. Veron, 2005) 2t3
G4(x)xt xt-1 1 t2 t (t2 t-1) t2 t-1-2l(t-3/2) 2t3
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• THANK YOU
• FOR YOUR ATTENTION!

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Code chain
Codes with odd minimal weight
Codes with even weight
G (x) xt-1 1
n22l-t1
G4(x) xt xt-1 1
G2(x) xt x L 2 L\0
n22l-t
n22l-t-1
G2(x) xt x L2 L2\a
G(x) xt1 1