Best Known (71, 71+21, s)-Nets in Base 3
(71, 71+21, 464)-Net over F3 — Constructive and digital
Digital (71, 92, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 23, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
(71, 71+21, 747)-Net over F3 — Digital
Digital (71, 92, 747)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(392, 747, F3, 21) (dual of [747, 655, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(392, 760, F3, 21) (dual of [760, 668, 22]-code), using
- construction XX applied to C1 = C([724,15]), C2 = C([1,16]), C3 = C1 + C2 = C([1,15]), and C∩ = C1 ∩ C2 = C([724,16]) [i] based on
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,15}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(366, 728, F3, 16) (dual of [728, 662, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(385, 728, F3, 21) (dual of [728, 643, 22]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,16}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(360, 728, F3, 15) (dual of [728, 668, 16]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [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
- construction XX applied to C1 = C([724,15]), C2 = C([1,16]), C3 = C1 + C2 = C([1,15]), and C∩ = C1 ∩ C2 = C([724,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(392, 760, F3, 21) (dual of [760, 668, 22]-code), using
(71, 71+21, 49735)-Net in Base 3 — Upper bound on s
There is no (71, 92, 49736)-net in base 3, because
- 1 times m-reduction [i] would yield (71, 91, 49736)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 26 186343 543529 649702 937810 598414 863365 449137 > 391 [i]