Best Known (31−11, 31, s)-Nets in Base 8
(31−11, 31, 208)-Net over F8 — Constructive and digital
Digital (20, 31, 208)-net over F8, using
- 3 times m-reduction [i] based on digital (20, 34, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 17, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 17, 104)-net over F64, using
(31−11, 31, 514)-Net in Base 8 — Constructive
(20, 31, 514)-net in base 8, using
- 1 times m-reduction [i] based on (20, 32, 514)-net in base 8, using
- base change [i] based on digital (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- base change [i] based on digital (12, 24, 514)-net over F16, using
(31−11, 31, 538)-Net over F8 — Digital
Digital (20, 31, 538)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(831, 538, F8, 11) (dual of [538, 507, 12]-code), using
- 18 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0) [i] based on linear OA(828, 517, F8, 11) (dual of [517, 489, 12]-code), using
- construction XX applied to C1 = C([510,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([510,9]) [i] based on
- linear OA(825, 511, F8, 10) (dual of [511, 486, 11]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(825, 511, F8, 10) (dual of [511, 486, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(828, 511, F8, 11) (dual of [511, 483, 12]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(822, 511, F8, 9) (dual of [511, 489, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [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([510,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([510,9]) [i] based on
- 18 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0) [i] based on linear OA(828, 517, F8, 11) (dual of [517, 489, 12]-code), using
(31−11, 31, 97558)-Net in Base 8 — Upper bound on s
There is no (20, 31, 97559)-net in base 8, because
- 1 times m-reduction [i] would yield (20, 30, 97559)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1237 958447 114431 192966 538570 > 830 [i]