Best Known (40−9, 40, s)-Nets in Base 3
(40−9, 40, 400)-Net over F3 — Constructive and digital
Digital (31, 40, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 10, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(40−9, 40, 751)-Net over F3 — Digital
Digital (31, 40, 751)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 751, F3, 9) (dual of [751, 711, 10]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0) [i] based on linear OA(337, 740, F3, 9) (dual of [740, 703, 10]-code), using
- construction XX applied to C1 = C([727,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([727,7]) [i] based on
- linear OA(331, 728, F3, 8) (dual of [728, 697, 9]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(331, 728, F3, 8) (dual of [728, 697, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(337, 728, F3, 9) (dual of [728, 691, 10]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(325, 728, F3, 7) (dual of [728, 703, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code) (see above)
- construction XX applied to C1 = C([727,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([727,7]) [i] based on
- 8 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0) [i] based on linear OA(337, 740, F3, 9) (dual of [740, 703, 10]-code), using
(40−9, 40, 49650)-Net in Base 3 — Upper bound on s
There is no (31, 40, 49651)-net in base 3, because
- 1 times m-reduction [i] would yield (31, 39, 49651)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 052688 179314 667481 > 339 [i]