Best Known (211−24, 211, s)-Nets in Base 4
(211−24, 211, 349534)-Net over F4 — Constructive and digital
Digital (187, 211, 349534)-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 (174, 198, 349525)-net over F4, using
- net defined by OOA [i] based on linear OOA(4198, 349525, F4, 24, 24) (dual of [(349525, 24), 8388402, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4198, 4194300, F4, 24) (dual of [4194300, 4194102, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4198, 4194304, F4, 24) (dual of [4194304, 4194106, 25]-code), using
- 1 times truncation [i] based on linear OA(4199, 4194305, F4, 25) (dual of [4194305, 4194106, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, 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(4199, 4194305, F4, 25) (dual of [4194305, 4194106, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4198, 4194304, F4, 24) (dual of [4194304, 4194106, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4198, 4194300, F4, 24) (dual of [4194300, 4194102, 25]-code), using
- net defined by OOA [i] based on linear OOA(4198, 349525, F4, 24, 24) (dual of [(349525, 24), 8388402, 25]-NRT-code), using
- digital (1, 13, 9)-net over F4, using
(211−24, 211, 2097185)-Net over F4 — Digital
Digital (187, 211, 2097185)-net over F4, using
- 41 times duplication [i] based on digital (186, 210, 2097185)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4210, 2097185, F4, 2, 24) (dual of [(2097185, 2), 4194160, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4210, 4194370, F4, 24) (dual of [4194370, 4194160, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(24) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4210, 4194370, F4, 24) (dual of [4194370, 4194160, 25]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4210, 2097185, F4, 2, 24) (dual of [(2097185, 2), 4194160, 25]-NRT-code), using
(211−24, 211, large)-Net in Base 4 — Upper bound on s
There is no (187, 211, large)-net in base 4, because
- 22 times m-reduction [i] would yield (187, 189, large)-net in base 4, but