Best Known (26, 26+12, s)-Nets in Base 8
(26, 26+12, 354)-Net over F8 — Constructive and digital
Digital (26, 38, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 19, 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
(26, 26+12, 523)-Net in Base 8 — Constructive
(26, 38, 523)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- (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
- digital (0, 6, 9)-net over F8, using
(26, 26+12, 931)-Net over F8 — Digital
Digital (26, 38, 931)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(838, 931, F8, 12) (dual of [931, 893, 13]-code), using
- 407 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0, 1, 37 times 0, 1, 75 times 0, 1, 117 times 0, 1, 154 times 0) [i] based on linear OA(831, 517, F8, 12) (dual of [517, 486, 13]-code), using
- construction XX applied to C1 = C([510,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([510,10]) [i] based on
- 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(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(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(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(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([510,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([510,10]) [i] based on
- 407 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0, 1, 37 times 0, 1, 75 times 0, 1, 117 times 0, 1, 154 times 0) [i] based on linear OA(831, 517, F8, 12) (dual of [517, 486, 13]-code), using
(26, 26+12, 224226)-Net in Base 8 — Upper bound on s
There is no (26, 38, 224227)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 20769 316060 890747 144043 546471 377400 > 838 [i]