Best Known (21−7, 21, s)-Nets in Base 8
(21−7, 21, 320)-Net over F8 — Constructive and digital
Digital (14, 21, 320)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 208)-net over F8, using
- net defined by OOA [i] based on linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(85, 208, F8, 2, 3) (dual of [(208, 2), 411, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- digital (9, 16, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 8, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 8, 80)-net over F64, using
- digital (2, 5, 208)-net over F8, using
(21−7, 21, 514)-Net in Base 8 — Constructive
(14, 21, 514)-net in base 8, using
- 1 times m-reduction [i] based on (14, 22, 514)-net in base 8, using
- trace code for nets [i] based on (3, 11, 257)-net in base 64, using
- 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
- base change [i] based on digital (0, 9, 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
- base change [i] based on digital (0, 9, 257)-net over F256, using
- 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
- trace code for nets [i] based on (3, 11, 257)-net in base 64, using
(21−7, 21, 651)-Net over F8 — Digital
Digital (14, 21, 651)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(821, 651, F8, 7) (dual of [651, 630, 8]-code), using
- 132 step Varšamov–Edel lengthening with (ri) = (1, 24 times 0, 1, 106 times 0) [i] based on linear OA(819, 517, F8, 7) (dual of [517, 498, 8]-code), using
- construction XX applied to C1 = C([510,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([510,5]) [i] based on
- linear OA(816, 511, F8, 6) (dual of [511, 495, 7]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [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(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(813, 511, F8, 5) (dual of [511, 498, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([510,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([510,5]) [i] based on
- 132 step Varšamov–Edel lengthening with (ri) = (1, 24 times 0, 1, 106 times 0) [i] based on linear OA(819, 517, F8, 7) (dual of [517, 498, 8]-code), using
(21−7, 21, 272197)-Net in Base 8 — Upper bound on s
There is no (14, 21, 272198)-net in base 8, because
- 1 times m-reduction [i] would yield (14, 20, 272198)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1 152934 133789 164406 > 820 [i]