Best Known (127−22, 127, s)-Nets in Base 4
(127−22, 127, 1504)-Net over F4 — Constructive and digital
Digital (105, 127, 1504)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 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 (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 (3, 14, 14)-net over F4, using
(127−22, 127, 16440)-Net over F4 — Digital
Digital (105, 127, 16440)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4127, 16440, F4, 22) (dual of [16440, 16313, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(13) [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(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(21) ⊂ Ce(13) [i] based on
(127−22, 127, large)-Net in Base 4 — Upper bound on s
There is no (105, 127, large)-net in base 4, because
- 20 times m-reduction [i] would yield (105, 107, large)-net in base 4, but