Best Known (40, 62, s)-Nets in Base 8
(40, 62, 354)-Net over F8 — Constructive and digital
Digital (40, 62, 354)-net over F8, using
- 4 times m-reduction [i] based on digital (40, 66, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 33, 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, 33, 177)-net over F64, using
(40, 62, 516)-Net in Base 8 — Constructive
(40, 62, 516)-net in base 8, using
- trace code for nets [i] based on (9, 31, 258)-net in base 64, using
- 1 times m-reduction [i] based on (9, 32, 258)-net in base 64, using
- base change [i] based on digital (1, 24, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 24, 258)-net over F256, using
- 1 times m-reduction [i] based on (9, 32, 258)-net in base 64, using
(40, 62, 590)-Net over F8 — Digital
Digital (40, 62, 590)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(862, 590, F8, 22) (dual of [590, 528, 23]-code), using
- 69 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 20 times 0, 1, 41 times 0) [i] based on linear OA(858, 517, F8, 22) (dual of [517, 459, 23]-code), using
- construction XX applied to C1 = C([510,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([510,20]) [i] based on
- linear OA(855, 511, F8, 21) (dual of [511, 456, 22]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(855, 511, F8, 21) (dual of [511, 456, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(858, 511, F8, 22) (dual of [511, 453, 23]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(852, 511, F8, 20) (dual of [511, 459, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [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([510,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([510,20]) [i] based on
- 69 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 20 times 0, 1, 41 times 0) [i] based on linear OA(858, 517, F8, 22) (dual of [517, 459, 23]-code), using
(40, 62, 86303)-Net in Base 8 — Upper bound on s
There is no (40, 62, 86304)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 98 090426 803482 733770 794453 078211 551558 954501 897243 359461 > 862 [i]