Best Known (135, 135+19, s)-Nets in Base 4
(135, 135+19, 116524)-Net over F4 — Constructive and digital
Digital (135, 154, 116524)-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 (122, 141, 116509)-net over F4, using
- net defined by OOA [i] based on linear OOA(4141, 116509, F4, 19, 19) (dual of [(116509, 19), 2213530, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4141, 1048582, F4, 19) (dual of [1048582, 1048441, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 1048586, F4, 19) (dual of [1048586, 1048445, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(4141, 1048586, F4, 19) (dual of [1048586, 1048445, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4141, 1048582, F4, 19) (dual of [1048582, 1048441, 20]-code), using
- net defined by OOA [i] based on linear OOA(4141, 116509, F4, 19, 19) (dual of [(116509, 19), 2213530, 20]-NRT-code), using
- digital (4, 13, 15)-net over F4, using
(135, 135+19, 627126)-Net over F4 — Digital
Digital (135, 154, 627126)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4154, 627126, F4, 19) (dual of [627126, 626972, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4154, 1048605, F4, 19) (dual of [1048605, 1048451, 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(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, 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(4154, 1048605, F4, 19) (dual of [1048605, 1048451, 20]-code), using
(135, 135+19, large)-Net in Base 4 — Upper bound on s
There is no (135, 154, large)-net in base 4, because
- 17 times m-reduction [i] would yield (135, 137, large)-net in base 4, but