Best Known (90, 90+27, s)-Nets in Base 3
(90, 90+27, 464)-Net over F3 — Constructive and digital
Digital (90, 117, 464)-net over F3, using
- 31 times duplication [i] based on digital (89, 116, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 29, 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
- trace code for nets [i] based on digital (2, 29, 116)-net over F81, using
(90, 90+27, 784)-Net over F3 — Digital
Digital (90, 117, 784)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3117, 784, F3, 27) (dual of [784, 667, 28]-code), using
- 30 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0) [i] based on linear OA(3111, 748, F3, 27) (dual of [748, 637, 28]-code), using
- construction XX applied to C1 = C([725,22]), C2 = C([0,24]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([725,24]) [i] based on
- linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,22}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(397, 728, F3, 25) (dual of [728, 631, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3109, 728, F3, 28) (dual of [728, 619, 29]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,24}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to C1 = C([725,22]), C2 = C([0,24]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([725,24]) [i] based on
- 30 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0) [i] based on linear OA(3111, 748, F3, 27) (dual of [748, 637, 28]-code), using
(90, 90+27, 51245)-Net in Base 3 — Upper bound on s
There is no (90, 117, 51246)-net in base 3, because
- 1 times m-reduction [i] would yield (90, 116, 51246)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 22 185340 395904 001310 460878 726943 283969 571852 864778 019557 > 3116 [i]