Best Known (109, 109+32, s)-Nets in Base 3
(109, 109+32, 464)-Net over F3 — Constructive and digital
Digital (109, 141, 464)-net over F3, using
- 31 times duplication [i] based on digital (108, 140, 464)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (107, 140, 464)-net over F3, using
(109, 109+32, 935)-Net over F3 — Digital
Digital (109, 141, 935)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 935, F3, 32) (dual of [935, 794, 33]-code), using
- 178 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0) [i] based on linear OA(3125, 741, F3, 32) (dual of [741, 616, 33]-code), using
- construction XX applied to C1 = C([334,364]), C2 = C([336,365]), C3 = C1 + C2 = C([336,364]), and C∩ = C1 ∩ C2 = C([334,365]) [i] based on
- linear OA(3118, 728, F3, 31) (dual of [728, 610, 32]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,364}, and designed minimum distance d ≥ |I|+1 = 32 [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 = {336,337,…,365}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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 = {334,335,…,365}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3112, 728, F3, 29) (dual of [728, 616, 30]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {336,337,…,364}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- 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, using
- 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([334,364]), C2 = C([336,365]), C3 = C1 + C2 = C([336,364]), and C∩ = C1 ∩ C2 = C([334,365]) [i] based on
- 178 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0) [i] based on linear OA(3125, 741, F3, 32) (dual of [741, 616, 33]-code), using
(109, 109+32, 54452)-Net in Base 3 — Upper bound on s
There is no (109, 141, 54453)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 18 801251 055973 123562 317827 109486 313817 154914 215689 315325 913195 597249 > 3141 [i]