Best Known (131−30, 131, s)-Nets in Base 3
(131−30, 131, 464)-Net over F3 — Constructive and digital
Digital (101, 131, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (101, 132, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 33, 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, 33, 116)-net over F81, using
(131−30, 131, 855)-Net over F3 — Digital
Digital (101, 131, 855)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3131, 855, F3, 30) (dual of [855, 724, 31]-code), using
- 105 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 20 times 0, 1, 24 times 0) [i] based on linear OA(3118, 737, F3, 30) (dual of [737, 619, 31]-code), using
- construction XX applied to C1 = C([727,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([727,28]) [i] based on
- linear OA(3115, 728, F3, 29) (dual of [728, 613, 30]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(3112, 728, F3, 29) (dual of [728, 616, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(3118, 728, F3, 30) (dual of [728, 610, 31]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3109, 728, F3, 28) (dual of [728, 619, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [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, 3, F3, 0) (dual of [3, 3, 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 (see above)
- construction XX applied to C1 = C([727,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([727,28]) [i] based on
- 105 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 20 times 0, 1, 24 times 0) [i] based on linear OA(3118, 737, F3, 30) (dual of [737, 619, 31]-code), using
(131−30, 131, 47147)-Net in Base 3 — Upper bound on s
There is no (101, 131, 47148)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 318 348156 323272 424290 345066 442303 579921 857749 523261 918844 099601 > 3131 [i]