Best Known (130−22, 130, s)-Nets in Base 5
(130−22, 130, 7106)-Net over F5 — Constructive and digital
Digital (108, 130, 7106)-net over F5, using
- 51 times duplication [i] based on digital (107, 129, 7106)-net over F5, using
- net defined by OOA [i] based on linear OOA(5129, 7106, F5, 22, 22) (dual of [(7106, 22), 156203, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5129, 78166, F5, 22) (dual of [78166, 78037, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5129, 78169, F5, 22) (dual of [78169, 78040, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5129, 78169, F5, 22) (dual of [78169, 78040, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(5129, 78166, F5, 22) (dual of [78166, 78037, 23]-code), using
- net defined by OOA [i] based on linear OOA(5129, 7106, F5, 22, 22) (dual of [(7106, 22), 156203, 23]-NRT-code), using
(130−22, 130, 66913)-Net over F5 — Digital
Digital (108, 130, 66913)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5130, 66913, F5, 22) (dual of [66913, 66783, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5130, 78170, F5, 22) (dual of [78170, 78040, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(510, 45, F5, 5) (dual of [45, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(510, 52, F5, 5) (dual of [52, 42, 6]-code), using
- trace code [i] based on linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- extended Reed–Solomon code RSe(21,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26 | 252−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- algebraic-geometric code AG(F,10P) with degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F, Q+6P) with degQ = 2 and degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- trace code [i] based on linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(510, 52, F5, 5) (dual of [52, 42, 6]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5130, 78170, F5, 22) (dual of [78170, 78040, 23]-code), using
(130−22, 130, large)-Net in Base 5 — Upper bound on s
There is no (108, 130, large)-net in base 5, because
- 20 times m-reduction [i] would yield (108, 110, large)-net in base 5, but