Best Known (41−13, 41, s)-Nets in Base 8
(41−13, 41, 354)-Net over F8 — Constructive and digital
Digital (28, 41, 354)-net over F8, using
- 1 times m-reduction [i] based on digital (28, 42, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 21, 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, 21, 177)-net over F64, using
(41−13, 41, 518)-Net in Base 8 — Constructive
(28, 41, 518)-net in base 8, using
- 81 times duplication [i] based on (27, 40, 518)-net in base 8, using
- base change [i] based on digital (17, 30, 518)-net over F16, using
- trace code for nets [i] based on digital (2, 15, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 15, 259)-net over F256, using
- base change [i] based on digital (17, 30, 518)-net over F16, using
(41−13, 41, 927)-Net over F8 — Digital
Digital (28, 41, 927)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(841, 927, F8, 13) (dual of [927, 886, 14]-code), using
- 403 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 17 times 0, 1, 42 times 0, 1, 79 times 0, 1, 112 times 0, 1, 143 times 0) [i] based on linear OA(834, 517, F8, 13) (dual of [517, 483, 14]-code), using
- construction XX applied to C1 = C([510,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([510,11]) [i] based on
- linear OA(831, 511, F8, 12) (dual of [511, 480, 13]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(831, 511, F8, 12) (dual of [511, 480, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(834, 511, F8, 13) (dual of [511, 477, 14]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(828, 511, F8, 11) (dual of [511, 483, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([510,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([510,11]) [i] based on
- 403 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 17 times 0, 1, 42 times 0, 1, 79 times 0, 1, 112 times 0, 1, 143 times 0) [i] based on linear OA(834, 517, F8, 13) (dual of [517, 483, 14]-code), using
(41−13, 41, 448456)-Net in Base 8 — Upper bound on s
There is no (28, 41, 448457)-net in base 8, because
- 1 times m-reduction [i] would yield (28, 40, 448457)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1 329229 876952 268667 328719 602764 543420 > 840 [i]