Best Known (231−41, 231, s)-Nets in Base 4
(231−41, 231, 1554)-Net over F4 — Constructive and digital
Digital (190, 231, 1554)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 24, 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 (166, 207, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 69, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 69, 513)-net over F64, using
- digital (4, 24, 15)-net over F4, using
(231−41, 231, 16452)-Net over F4 — Digital
Digital (190, 231, 16452)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4231, 16452, F4, 41) (dual of [16452, 16221, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(30) [i] based on
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(420, 68, F4, 9) (dual of [68, 48, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(419, 64, F4, 9) (dual of [64, 45, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- construction X applied to Ce(40) ⊂ Ce(30) [i] based on
(231−41, 231, large)-Net in Base 4 — Upper bound on s
There is no (190, 231, large)-net in base 4, because
- 39 times m-reduction [i] would yield (190, 192, large)-net in base 4, but