Best Known (141−33, 141, s)-Nets in Base 3
(141−33, 141, 464)-Net over F3 — Constructive and digital
Digital (108, 141, 464)-net over F3, using
- 31 times duplication [i] based on digital (107, 140, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 35, 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, 35, 116)-net over F81, using
(141−33, 141, 832)-Net over F3 — Digital
Digital (108, 141, 832)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 832, F3, 33) (dual of [832, 691, 34]-code), using
- 81 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 17 times 0, 1, 20 times 0) [i] based on linear OA(3130, 740, F3, 33) (dual of [740, 610, 34]-code), using
- construction XX applied to C1 = C([727,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([727,31]) [i] based on
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3130, 728, F3, 33) (dual of [728, 598, 34]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,31}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3118, 728, F3, 31) (dual of [728, 610, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [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([727,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([727,31]) [i] based on
- 81 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 17 times 0, 1, 20 times 0) [i] based on linear OA(3130, 740, F3, 33) (dual of [740, 610, 34]-code), using
(141−33, 141, 50837)-Net in Base 3 — Upper bound on s
There is no (108, 141, 50838)-net in base 3, because
- 1 times m-reduction [i] would yield (108, 140, 50838)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 266046 959317 783169 189882 169908 242066 546667 991769 368290 601802 802337 > 3140 [i]