Best Known (77−29, 77, s)-Nets in Base 8
(77−29, 77, 354)-Net over F8 — Constructive and digital
Digital (48, 77, 354)-net over F8, using
- 5 times m-reduction [i] based on digital (48, 82, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
(77−29, 77, 384)-Net in Base 8 — Constructive
(48, 77, 384)-net in base 8, using
- 1 times m-reduction [i] based on (48, 78, 384)-net in base 8, using
- trace code for nets [i] based on (9, 39, 192)-net in base 64, using
- 3 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- 3 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- trace code for nets [i] based on (9, 39, 192)-net in base 64, using
(77−29, 77, 524)-Net over F8 — Digital
Digital (48, 77, 524)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(877, 524, F8, 29) (dual of [524, 447, 30]-code), using
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(876, 517, F8, 29) (dual of [517, 441, 30]-code), using
- construction XX applied to C1 = C([510,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([510,27]) [i] based on
- linear OA(873, 511, F8, 28) (dual of [511, 438, 29]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,26}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(873, 511, F8, 28) (dual of [511, 438, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(876, 511, F8, 29) (dual of [511, 435, 30]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(870, 511, F8, 27) (dual of [511, 441, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 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
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([510,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([510,27]) [i] based on
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(876, 517, F8, 29) (dual of [517, 441, 30]-code), using
(77−29, 77, 68991)-Net in Base 8 — Upper bound on s
There is no (48, 77, 68992)-net in base 8, because
- 1 times m-reduction [i] would yield (48, 76, 68992)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 431 415264 891803 028761 833794 698591 309548 629505 725698 997198 675311 310929 > 876 [i]