Best Known (126−19, 126, s)-Nets in Base 4
(126−19, 126, 7297)-Net over F4 — Constructive and digital
Digital (107, 126, 7297)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (94, 113, 7282)-net over F4, using
- net defined by OOA [i] based on linear OOA(4113, 7282, F4, 19, 19) (dual of [(7282, 19), 138245, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4113, 65539, F4, 19) (dual of [65539, 65426, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4113, 65544, F4, 19) (dual of [65544, 65431, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4113, 65536, F4, 19) (dual of [65536, 65423, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4113, 65544, F4, 19) (dual of [65544, 65431, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4113, 65539, F4, 19) (dual of [65539, 65426, 20]-code), using
- net defined by OOA [i] based on linear OOA(4113, 7282, F4, 19, 19) (dual of [(7282, 19), 138245, 20]-NRT-code), using
- digital (4, 13, 15)-net over F4, using
(126−19, 126, 63921)-Net over F4 — Digital
Digital (107, 126, 63921)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4126, 63921, F4, 19) (dual of [63921, 63795, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 19) (dual of [65565, 65439, 20]-code), using
- (u, u+v)-construction [i] based on
- linear OA(413, 28, F4, 9) (dual of [28, 15, 10]-code), using
- 2 times truncation [i] based on linear OA(415, 30, F4, 11) (dual of [30, 15, 12]-code), using
- extended quadratic residue code Qe(30,4) [i]
- 2 times truncation [i] based on linear OA(415, 30, F4, 11) (dual of [30, 15, 12]-code), using
- linear OA(4113, 65537, F4, 19) (dual of [65537, 65424, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(413, 28, F4, 9) (dual of [28, 15, 10]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 19) (dual of [65565, 65439, 20]-code), using
(126−19, 126, large)-Net in Base 4 — Upper bound on s
There is no (107, 126, large)-net in base 4, because
- 17 times m-reduction [i] would yield (107, 109, large)-net in base 4, but