Best Known (186, 186+34, s)-Nets in Base 4
(186, 186+34, 3865)-Net over F4 — Constructive and digital
Digital (186, 220, 3865)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 19, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (167, 201, 3855)-net over F4, using
- net defined by OOA [i] based on linear OOA(4201, 3855, F4, 34, 34) (dual of [(3855, 34), 130869, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4201, 65535, F4, 34) (dual of [65535, 65334, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- discarding factors / shortening the dual code based on linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4201, 65535, F4, 34) (dual of [65535, 65334, 35]-code), using
- net defined by OOA [i] based on linear OOA(4201, 3855, F4, 34, 34) (dual of [(3855, 34), 130869, 35]-NRT-code), using
- digital (2, 19, 10)-net over F4, using
(186, 186+34, 56222)-Net over F4 — Digital
Digital (186, 220, 56222)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4220, 56222, F4, 34) (dual of [56222, 56002, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4220, 65605, F4, 34) (dual of [65605, 65385, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(24) [i] based on
- linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(419, 69, F4, 8) (dual of [69, 50, 9]-code), using
- construction XX applied to C1 = C({0,1,2,3,5,47}), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C({0,1,2,3,5,6,47}) [i] based on
- linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,5,47}, and minimum distance d ≥ |{−1,0,…,5}|+1 = 8 (BCH-bound) [i]
- linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(419, 63, F4, 8) (dual of [63, 44, 9]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,5,6,47}, and minimum distance d ≥ |{−1,0,…,6}|+1 = 9 (BCH-bound) [i]
- linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 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
- linear OA(40, 3, F4, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C({0,1,2,3,5,47}), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C({0,1,2,3,5,6,47}) [i] based on
- construction X applied to Ce(33) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4220, 65605, F4, 34) (dual of [65605, 65385, 35]-code), using
(186, 186+34, large)-Net in Base 4 — Upper bound on s
There is no (186, 220, large)-net in base 4, because
- 32 times m-reduction [i] would yield (186, 188, large)-net in base 4, but