Best Known (15, 22, s)-Nets in Base 8
(15, 22, 333)-Net over F8 — Constructive and digital
Digital (15, 22, 333)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 3, 73)-net over F8, using
- digital (2, 5, 130)-net over F8, using
- s-reduction 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
- s-reduction based on digital (2, 5, 208)-net over F8, using
- digital (7, 14, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 65)-net over F64, using
(15, 22, 516)-Net in Base 8 — Constructive
(15, 22, 516)-net in base 8, using
- trace code for nets [i] based on (4, 11, 258)-net in base 64, using
- 1 times m-reduction [i] based on (4, 12, 258)-net in base 64, using
- base change [i] based on digital (1, 9, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 9, 258)-net over F256, using
- 1 times m-reduction [i] based on (4, 12, 258)-net in base 64, using
(15, 22, 884)-Net over F8 — Digital
Digital (15, 22, 884)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(822, 884, F8, 7) (dual of [884, 862, 8]-code), using
- 364 step Varšamov–Edel lengthening with (ri) = (1, 24 times 0, 1, 106 times 0, 1, 231 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
- 364 step Varšamov–Edel lengthening with (ri) = (1, 24 times 0, 1, 106 times 0, 1, 231 times 0) [i] based on linear OA(819, 517, F8, 7) (dual of [517, 498, 8]-code), using
(15, 22, 544395)-Net in Base 8 — Upper bound on s
There is no (15, 22, 544396)-net in base 8, because
- 1 times m-reduction [i] would yield (15, 21, 544396)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 9 223400 460056 071227 > 821 [i]