Best Known (49−16, 49, s)-Nets in Base 9
(49−16, 49, 344)-Net over F9 — Constructive and digital
Digital (33, 49, 344)-net over F9, using
- 3 times m-reduction [i] based on digital (33, 52, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 26, 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, 26, 172)-net over F81, using
(49−16, 49, 1061)-Net over F9 — Digital
Digital (33, 49, 1061)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(949, 1061, F9, 16) (dual of [1061, 1012, 17]-code), using
- 321 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 22 times 0, 1, 56 times 0, 1, 98 times 0, 1, 135 times 0) [i] based on linear OA(943, 734, F9, 16) (dual of [734, 691, 17]-code), using
- construction XX applied to C1 = C([727,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([727,14]) [i] based on
- linear OA(940, 728, F9, 15) (dual of [728, 688, 16]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(940, 728, F9, 15) (dual of [728, 688, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(943, 728, F9, 16) (dual of [728, 685, 17]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(937, 728, F9, 14) (dual of [728, 691, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [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([727,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([727,14]) [i] based on
- 321 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 22 times 0, 1, 56 times 0, 1, 98 times 0, 1, 135 times 0) [i] based on linear OA(943, 734, F9, 16) (dual of [734, 691, 17]-code), using
(49−16, 49, 329101)-Net in Base 9 — Upper bound on s
There is no (33, 49, 329102)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 57265 235543 052856 242521 063426 387007 917065 241217 > 949 [i]