Best Known (127, 127+23, s)-Nets in Base 4
(127, 127+23, 5968)-Net over F4 — Constructive and digital
Digital (127, 150, 5968)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 13, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (114, 137, 5958)-net over F4, using
- net defined by OOA [i] based on linear OOA(4137, 5958, F4, 23, 23) (dual of [(5958, 23), 136897, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4137, 65539, F4, 23) (dual of [65539, 65402, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4137, 65544, F4, 23) (dual of [65544, 65407, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4137, 65544, F4, 23) (dual of [65544, 65407, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4137, 65539, F4, 23) (dual of [65539, 65402, 24]-code), using
- net defined by OOA [i] based on linear OOA(4137, 5958, F4, 23, 23) (dual of [(5958, 23), 136897, 24]-NRT-code), using
- digital (2, 13, 10)-net over F4, using
(127, 127+23, 54075)-Net over F4 — Digital
Digital (127, 150, 54075)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4150, 54075, F4, 23) (dual of [54075, 53925, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4150, 65553, F4, 23) (dual of [65553, 65403, 24]-code), using
- (u, u+v)-construction [i] based on
- linear OA(413, 17, F4, 11) (dual of [17, 4, 12]-code), using
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4150, 65553, F4, 23) (dual of [65553, 65403, 24]-code), using
(127, 127+23, large)-Net in Base 4 — Upper bound on s
There is no (127, 150, large)-net in base 4, because
- 21 times m-reduction [i] would yield (127, 129, large)-net in base 4, but