Best Known (149, 149+19, s)-Nets in Base 4
(149, 149+19, 466050)-Net over F4 — Constructive and digital
Digital (149, 168, 466050)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 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 (137, 156, 466036)-net over F4, using
- net defined by OOA [i] based on linear OOA(4156, 466036, F4, 19, 19) (dual of [(466036, 19), 8854528, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4156, 4194325, F4, 19) (dual of [4194325, 4194169, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4156, 4194328, F4, 19) (dual of [4194328, 4194172, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(4155, 4194305, F4, 19) (dual of [4194305, 4194150, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 23, F4, 1) (dual of [23, 22, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4156, 4194328, F4, 19) (dual of [4194328, 4194172, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4156, 4194325, F4, 19) (dual of [4194325, 4194169, 20]-code), using
- net defined by OOA [i] based on linear OOA(4156, 466036, F4, 19, 19) (dual of [(466036, 19), 8854528, 20]-NRT-code), using
- digital (3, 12, 14)-net over F4, using
(149, 149+19, 2097186)-Net over F4 — Digital
Digital (149, 168, 2097186)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4168, 2097186, F4, 2, 19) (dual of [(2097186, 2), 4194204, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4166, 2097185, F4, 2, 19) (dual of [(2097185, 2), 4194204, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4166, 4194370, F4, 19) (dual of [4194370, 4194204, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(4155, 4194304, F4, 19) (dual of [4194304, 4194149, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4100, 4194304, F4, 13) (dual of [4194304, 4194204, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(18) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(4166, 4194370, F4, 19) (dual of [4194370, 4194204, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4166, 2097185, F4, 2, 19) (dual of [(2097185, 2), 4194204, 20]-NRT-code), using
(149, 149+19, large)-Net in Base 4 — Upper bound on s
There is no (149, 168, large)-net in base 4, because
- 17 times m-reduction [i] would yield (149, 151, large)-net in base 4, but