Best Known (24−8, 24, s)-Nets in Base 8
(24−8, 24, 260)-Net over F8 — Constructive and digital
Digital (16, 24, 260)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 4, 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, 4, 65)-net over F64, using
- digital (8, 16, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 8, 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, 8, 65)-net over F64, using
- digital (4, 8, 130)-net over F8, using
(24−8, 24, 516)-Net in Base 8 — Constructive
(16, 24, 516)-net in base 8, using
- base change [i] based on digital (10, 18, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 9, 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, 9, 258)-net over F256, using
(24−8, 24, 630)-Net over F8 — Digital
Digital (16, 24, 630)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(824, 630, F8, 8) (dual of [630, 606, 9]-code), using
- 111 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0, 1, 89 times 0) [i] based on linear OA(822, 517, F8, 8) (dual of [517, 495, 9]-code), using
- construction XX applied to C1 = C([510,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([510,6]) [i] based on
- linear OA(819, 511, F8, 7) (dual of [511, 492, 8]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(819, 511, F8, 7) (dual of [511, 492, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(822, 511, F8, 8) (dual of [511, 489, 9]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(816, 511, F8, 6) (dual of [511, 495, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([510,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([510,6]) [i] based on
- 111 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0, 1, 89 times 0) [i] based on linear OA(822, 517, F8, 8) (dual of [517, 495, 9]-code), using
(24−8, 24, 82886)-Net in Base 8 — Upper bound on s
There is no (16, 24, 82887)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 4722 478786 318602 139167 > 824 [i]