Best Known (177, 200, s)-Nets in Base 4
(177, 200, 381310)-Net over F4 — Constructive and digital
Digital (177, 200, 381310)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 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 (165, 188, 381301)-net over F4, using
- net defined by OOA [i] based on linear OOA(4188, 381301, F4, 23, 23) (dual of [(381301, 23), 8769735, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4188, 4194312, F4, 23) (dual of [4194312, 4194124, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4188, 4194315, F4, 23) (dual of [4194315, 4194127, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4188, 4194315, F4, 23) (dual of [4194315, 4194127, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4188, 4194312, F4, 23) (dual of [4194312, 4194124, 24]-code), using
- net defined by OOA [i] based on linear OOA(4188, 381301, F4, 23, 23) (dual of [(381301, 23), 8769735, 24]-NRT-code), using
- digital (1, 12, 9)-net over F4, using
(177, 200, 2097185)-Net over F4 — Digital
Digital (177, 200, 2097185)-net over F4, using
- 41 times duplication [i] based on digital (176, 199, 2097185)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4199, 2097185, F4, 2, 23) (dual of [(2097185, 2), 4194171, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4199, 4194370, F4, 23) (dual of [4194370, 4194171, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(22) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4199, 4194370, F4, 23) (dual of [4194370, 4194171, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4199, 2097185, F4, 2, 23) (dual of [(2097185, 2), 4194171, 24]-NRT-code), using
(177, 200, large)-Net in Base 4 — Upper bound on s
There is no (177, 200, large)-net in base 4, because
- 21 times m-reduction [i] would yield (177, 179, large)-net in base 4, but