Best Known (201−31, 201, s)-Nets in Base 4
(201−31, 201, 4378)-Net over F4 — Constructive and digital
Digital (170, 201, 4378)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 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 (154, 185, 4369)-net over F4, using
- net defined by OOA [i] based on linear OOA(4185, 4369, F4, 31, 31) (dual of [(4369, 31), 135254, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using
- net defined by OOA [i] based on linear OOA(4185, 4369, F4, 31, 31) (dual of [(4369, 31), 135254, 32]-NRT-code), using
- digital (1, 16, 9)-net over F4, using
(201−31, 201, 55202)-Net over F4 — Digital
Digital (170, 201, 55202)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4201, 55202, F4, 31) (dual of [55202, 55001, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 65600, F4, 31) (dual of [65600, 65399, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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 Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4201, 65600, F4, 31) (dual of [65600, 65399, 32]-code), using
(201−31, 201, large)-Net in Base 4 — Upper bound on s
There is no (170, 201, large)-net in base 4, because
- 29 times m-reduction [i] would yield (170, 172, large)-net in base 4, but