Best Known (44−13, 44, s)-Nets in Base 4
(44−13, 44, 240)-Net over F4 — Constructive and digital
Digital (31, 44, 240)-net over F4, using
- 1 times m-reduction [i] based on digital (31, 45, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
(44−13, 44, 305)-Net over F4 — Digital
Digital (31, 44, 305)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(444, 305, F4, 13) (dual of [305, 261, 14]-code), using
- 38 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0) [i] based on linear OA(438, 261, F4, 13) (dual of [261, 223, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(437, 256, F4, 13) (dual of [256, 219, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(433, 256, F4, 11) (dual of [256, 223, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(41, 5, F4, 1) (dual of [5, 4, 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(12) ⊂ Ce(10) [i] based on
- 38 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0) [i] based on linear OA(438, 261, F4, 13) (dual of [261, 223, 14]-code), using
(44−13, 44, 20595)-Net in Base 4 — Upper bound on s
There is no (31, 44, 20596)-net in base 4, because
- 1 times m-reduction [i] would yield (31, 43, 20596)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 77 385848 754068 248962 302884 > 443 [i]