Best Known (38, 38+14, s)-Nets in Base 8
(38, 38+14, 587)-Net over F8 — Constructive and digital
Digital (38, 52, 587)-net over F8, using
- net defined by OOA [i] based on linear OOA(852, 587, F8, 14, 14) (dual of [(587, 14), 8166, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(852, 4109, F8, 14) (dual of [4109, 4057, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(852, 4111, F8, 14) (dual of [4111, 4059, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(849, 4096, F8, 14) (dual of [4096, 4047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(83, 15, F8, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(852, 4111, F8, 14) (dual of [4111, 4059, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(852, 4109, F8, 14) (dual of [4109, 4057, 15]-code), using
(38, 38+14, 644)-Net in Base 8 — Constructive
(38, 52, 644)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 7, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 7, 65)-net over F64, using
- (24, 38, 514)-net in base 8, using
- trace code for nets [i] based on (5, 19, 257)-net in base 64, using
- 1 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- base change [i] based on digital (0, 15, 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
- base change [i] based on digital (0, 15, 257)-net over F256, using
- 1 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- trace code for nets [i] based on (5, 19, 257)-net in base 64, using
- digital (7, 14, 130)-net over F8, using
(38, 38+14, 4157)-Net over F8 — Digital
Digital (38, 52, 4157)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(852, 4157, F8, 14) (dual of [4157, 4105, 15]-code), using
- 54 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 46 times 0) [i] based on linear OA(849, 4100, F8, 14) (dual of [4100, 4051, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(849, 4096, F8, 14) (dual of [4096, 4047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 54 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 46 times 0) [i] based on linear OA(849, 4100, F8, 14) (dual of [4100, 4051, 15]-code), using
(38, 38+14, 2468814)-Net in Base 8 — Upper bound on s
There is no (38, 52, 2468815)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 91344 099986 855453 248623 818553 018604 767387 903872 > 852 [i]