Best Known (178−20, 178, s)-Nets in Base 4
(178−20, 178, 419444)-Net over F4 — Constructive and digital
Digital (158, 178, 419444)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (145, 165, 419430)-net over F4, using
- net defined by OOA [i] based on linear OOA(4165, 419430, F4, 20, 20) (dual of [(419430, 20), 8388435, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4165, 4194300, F4, 20) (dual of [4194300, 4194135, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4165, 4194303, F4, 20) (dual of [4194303, 4194138, 21]-code), using
- 1 times truncation [i] based on linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times truncation [i] based on linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4165, 4194303, F4, 20) (dual of [4194303, 4194138, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4165, 4194300, F4, 20) (dual of [4194300, 4194135, 21]-code), using
- net defined by OOA [i] based on linear OOA(4165, 419430, F4, 20, 20) (dual of [(419430, 20), 8388435, 21]-NRT-code), using
- digital (3, 13, 14)-net over F4, using
(178−20, 178, 2097185)-Net over F4 — Digital
Digital (158, 178, 2097185)-net over F4, using
- 41 times duplication [i] based on digital (157, 177, 2097185)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4177, 2097185, F4, 2, 20) (dual of [(2097185, 2), 4194193, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4177, 4194370, F4, 20) (dual of [4194370, 4194193, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4111, 4194304, F4, 14) (dual of [4194304, 4194193, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(411, 66, F4, 5) (dual of [66, 55, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(47, 64, F4, 3) (dual of [64, 57, 4]-code or 64-cap in PG(6,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 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(4) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(4177, 4194370, F4, 20) (dual of [4194370, 4194193, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4177, 2097185, F4, 2, 20) (dual of [(2097185, 2), 4194193, 21]-NRT-code), using
(178−20, 178, large)-Net in Base 4 — Upper bound on s
There is no (158, 178, large)-net in base 4, because
- 18 times m-reduction [i] would yield (158, 160, large)-net in base 4, but