Best Known (234−36, 234, s)-Nets in Base 4
(234−36, 234, 3645)-Net over F4 — Constructive and digital
Digital (198, 234, 3645)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 18, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (180, 216, 3640)-net over F4, using
- net defined by OOA [i] based on linear OOA(4216, 3640, F4, 36, 36) (dual of [(3640, 36), 130824, 37]-NRT-code), using
- OA 18-folding and stacking [i] based on linear OA(4216, 65520, F4, 36) (dual of [65520, 65304, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4216, 65535, F4, 36) (dual of [65535, 65319, 37]-code), using
- 1 times truncation [i] based on linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 1 times truncation [i] based on linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4216, 65535, F4, 36) (dual of [65535, 65319, 37]-code), using
- OA 18-folding and stacking [i] based on linear OA(4216, 65520, F4, 36) (dual of [65520, 65304, 37]-code), using
- net defined by OOA [i] based on linear OOA(4216, 3640, F4, 36, 36) (dual of [(3640, 36), 130824, 37]-NRT-code), using
- digital (0, 18, 5)-net over F4, using
(234−36, 234, 60261)-Net over F4 — Digital
Digital (198, 234, 60261)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4234, 60261, F4, 36) (dual of [60261, 60027, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4234, 65601, F4, 36) (dual of [65601, 65367, 37]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4230, 65597, F4, 36) (dual of [65597, 65367, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(413, 61, F4, 6) (dual of [61, 48, 7]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4230, 65597, F4, 36) (dual of [65597, 65367, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4234, 65601, F4, 36) (dual of [65601, 65367, 37]-code), using
(234−36, 234, large)-Net in Base 4 — Upper bound on s
There is no (198, 234, large)-net in base 4, because
- 34 times m-reduction [i] would yield (198, 200, large)-net in base 4, but