Best Known (116, 116+36, s)-Nets in Base 3
(116, 116+36, 464)-Net over F3 — Constructive and digital
Digital (116, 152, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 38, 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
(116, 116+36, 840)-Net over F3 — Digital
Digital (116, 152, 840)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3152, 840, F3, 36) (dual of [840, 688, 37]-code), using
- 90 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 19 times 0, 1, 22 times 0) [i] based on linear OA(3142, 740, F3, 36) (dual of [740, 598, 37]-code), using
- construction XX applied to C1 = C([727,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([727,34]) [i] based on
- linear OA(3136, 728, F3, 35) (dual of [728, 592, 36]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3136, 728, F3, 35) (dual of [728, 592, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3142, 728, F3, 36) (dual of [728, 586, 37]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3130, 728, F3, 34) (dual of [728, 598, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [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([727,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([727,34]) [i] based on
- 90 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 19 times 0, 1, 22 times 0) [i] based on linear OA(3142, 740, F3, 36) (dual of [740, 598, 37]-code), using
(116, 116+36, 40358)-Net in Base 3 — Upper bound on s
There is no (116, 152, 40359)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 330537 621691 077258 649163 328645 610832 520340 458720 946560 999995 364324 922285 > 3152 [i]