Best Known (204, 204+37, s)-Nets in Base 4
(204, 204+37, 3658)-Net over F4 — Constructive and digital
Digital (204, 241, 3658)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 23, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (181, 218, 3641)-net over F4, using
- net defined by OOA [i] based on linear OOA(4218, 3641, F4, 37, 37) (dual of [(3641, 37), 134499, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4218, 65539, F4, 37) (dual of [65539, 65321, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 65545, F4, 37) (dual of [65545, 65327, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 65545, F4, 37) (dual of [65545, 65327, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4218, 65539, F4, 37) (dual of [65539, 65321, 38]-code), using
- net defined by OOA [i] based on linear OOA(4218, 3641, F4, 37, 37) (dual of [(3641, 37), 134499, 38]-NRT-code), using
- digital (5, 23, 17)-net over F4, using
(204, 204+37, 62281)-Net over F4 — Digital
Digital (204, 241, 62281)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4241, 62281, F4, 37) (dual of [62281, 62040, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4241, 65601, F4, 37) (dual of [65601, 65360, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,14]) [i] based on
- linear OA(4225, 65537, F4, 37) (dual of [65537, 65312, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(4177, 65537, F4, 29) (dual of [65537, 65360, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to C([0,18]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4241, 65601, F4, 37) (dual of [65601, 65360, 38]-code), using
(204, 204+37, large)-Net in Base 4 — Upper bound on s
There is no (204, 241, large)-net in base 4, because
- 35 times m-reduction [i] would yield (204, 206, large)-net in base 4, but