Best Known (124−22, 124, s)-Nets in Base 4
(124−22, 124, 1495)-Net over F4 — Constructive and digital
Digital (102, 124, 1495)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (91, 113, 1490)-net over F4, using
- net defined by OOA [i] based on linear OOA(4113, 1490, F4, 22, 22) (dual of [(1490, 22), 32667, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4113, 16390, F4, 22) (dual of [16390, 16277, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4113, 16391, F4, 22) (dual of [16391, 16278, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4113, 16391, F4, 22) (dual of [16391, 16278, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4113, 16390, F4, 22) (dual of [16390, 16277, 23]-code), using
- net defined by OOA [i] based on linear OOA(4113, 1490, F4, 22, 22) (dual of [(1490, 22), 32667, 23]-NRT-code), using
- digital (0, 11, 5)-net over F4, using
(124−22, 124, 13943)-Net over F4 — Digital
Digital (102, 124, 13943)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4124, 13943, F4, 22) (dual of [13943, 13819, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4124, 16396, F4, 22) (dual of [16396, 16272, 23]-code), using
- (u, u+v)-construction [i] based on
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4124, 16396, F4, 22) (dual of [16396, 16272, 23]-code), using
(124−22, 124, large)-Net in Base 4 — Upper bound on s
There is no (102, 124, large)-net in base 4, because
- 20 times m-reduction [i] would yield (102, 104, large)-net in base 4, but