Best Known (38, 59, s)-Nets in Base 9
(38, 59, 344)-Net over F9 — Constructive and digital
Digital (38, 59, 344)-net over F9, using
- 3 times m-reduction [i] based on digital (38, 62, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
(38, 59, 759)-Net over F9 — Digital
Digital (38, 59, 759)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(959, 759, F9, 21) (dual of [759, 700, 22]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 13 times 0) [i] based on linear OA(955, 734, F9, 21) (dual of [734, 679, 22]-code), using
- construction XX applied to C1 = C([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- linear OA(952, 728, F9, 20) (dual of [728, 676, 21]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(952, 728, F9, 20) (dual of [728, 676, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(955, 728, F9, 21) (dual of [728, 673, 22]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(949, 728, F9, 19) (dual of [728, 679, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 13 times 0) [i] based on linear OA(955, 734, F9, 21) (dual of [734, 679, 22]-code), using
(38, 59, 193856)-Net in Base 9 — Upper bound on s
There is no (38, 59, 193857)-net in base 9, because
- 1 times m-reduction [i] would yield (38, 58, 193857)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 22 186077 525857 839033 470431 152729 564411 280607 832990 898065 > 958 [i]