Best Known (193−24, 193, s)-Nets in Base 4
(193−24, 193, 87390)-Net over F4 — Constructive and digital
Digital (169, 193, 87390)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (156, 180, 87381)-net over F4, using
- net defined by OOA [i] based on linear OOA(4180, 87381, F4, 24, 24) (dual of [(87381, 24), 2096964, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4180, 1048572, F4, 24) (dual of [1048572, 1048392, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4180, 1048576, F4, 24) (dual of [1048576, 1048396, 25]-code), using
- 1 times truncation [i] based on linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4180, 1048576, F4, 24) (dual of [1048576, 1048396, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4180, 1048572, F4, 24) (dual of [1048572, 1048392, 25]-code), using
- net defined by OOA [i] based on linear OOA(4180, 87381, F4, 24, 24) (dual of [(87381, 24), 2096964, 25]-NRT-code), using
- digital (1, 13, 9)-net over F4, using
(193−24, 193, 542075)-Net over F4 — Digital
Digital (169, 193, 542075)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4193, 542075, F4, 24) (dual of [542075, 541882, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, 1048638, F4, 24) (dual of [1048638, 1048445, 25]-code), using
- 1 times truncation [i] based on linear OA(4194, 1048639, F4, 25) (dual of [1048639, 1048445, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- 1 times truncation [i] based on linear OA(4194, 1048639, F4, 25) (dual of [1048639, 1048445, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, 1048638, F4, 24) (dual of [1048638, 1048445, 25]-code), using
(193−24, 193, large)-Net in Base 4 — Upper bound on s
There is no (169, 193, large)-net in base 4, because
- 22 times m-reduction [i] would yield (169, 171, large)-net in base 4, but