Best Known (24, 24+12, s)-Nets in Base 8
(24, 24+12, 260)-Net over F8 — Constructive and digital
Digital (24, 36, 260)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
- digital (12, 24, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 12, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- trace code for nets [i] based on digital (0, 12, 65)-net over F64, using
- digital (6, 12, 130)-net over F8, using
(24, 24+12, 516)-Net in Base 8 — Constructive
(24, 36, 516)-net in base 8, using
- base change [i] based on digital (15, 27, 516)-net over F16, using
- 1 times m-reduction [i] based on digital (15, 28, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 14, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 14, 258)-net over F256, using
- 1 times m-reduction [i] based on digital (15, 28, 516)-net over F16, using
(24, 24+12, 656)-Net over F8 — Digital
Digital (24, 36, 656)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(836, 656, F8, 12) (dual of [656, 620, 13]-code), using
- 134 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0, 1, 37 times 0, 1, 75 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
- 134 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0, 1, 37 times 0, 1, 75 times 0) [i] based on linear OA(831, 517, F8, 12) (dual of [517, 486, 13]-code), using
(24, 24+12, 112111)-Net in Base 8 — Upper bound on s
There is no (24, 36, 112112)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 324 523664 290204 459593 477417 657205 > 836 [i]