Best Known (38, 38+10, s)-Nets in Base 4
(38, 38+10, 1054)-Net over F4 — Constructive and digital
Digital (38, 48, 1054)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 8, 26)-net over F4, using
- digital (30, 40, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 10, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 10, 257)-net over F256, using
(38, 38+10, 4146)-Net over F4 — Digital
Digital (38, 48, 4146)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(448, 4146, F4, 10) (dual of [4146, 4098, 11]-code), using
- 39 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 21 times 0) [i] based on linear OA(443, 4102, F4, 10) (dual of [4102, 4059, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(443, 4096, F4, 10) (dual of [4096, 4053, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(437, 4096, F4, 9) (dual of [4096, 4059, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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(9) ⊂ Ce(8) [i] based on
- 39 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 21 times 0) [i] based on linear OA(443, 4102, F4, 10) (dual of [4102, 4059, 11]-code), using
(38, 38+10, 522983)-Net in Base 4 — Upper bound on s
There is no (38, 48, 522984)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 79228 651436 942641 979782 715587 > 448 [i]