Best Known (111, 143, s)-Nets in Base 3
(111, 143, 464)-Net over F3 — Constructive and digital
Digital (111, 143, 464)-net over F3, using
- t-expansion [i] based on digital (110, 143, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (110, 144, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 36, 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, 36, 116)-net over F81, using
- 1 times m-reduction [i] based on digital (110, 144, 464)-net over F3, using
(111, 143, 1002)-Net over F3 — Digital
Digital (111, 143, 1002)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3143, 1002, F3, 32) (dual of [1002, 859, 33]-code), using
- 243 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, 1, 31 times 0, 1, 32 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
- 243 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, 1, 31 times 0, 1, 32 times 0) [i] based on linear OA(3125, 741, F3, 32) (dual of [741, 616, 33]-code), using
(111, 143, 62470)-Net in Base 3 — Upper bound on s
There is no (111, 143, 62471)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 169 215192 152797 955341 359954 503246 186478 329793 634911 671218 806423 288033 > 3143 [i]