Best Known (78, 78+25, s)-Nets in Base 3
(78, 78+25, 328)-Net over F3 — Constructive and digital
Digital (78, 103, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (78, 104, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 26, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 26, 82)-net over F81, using
(78, 78+25, 596)-Net over F3 — Digital
Digital (78, 103, 596)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3103, 596, F3, 25) (dual of [596, 493, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3103, 753, F3, 25) (dual of [753, 650, 26]-code), using
- construction XX applied to C1 = C([725,19]), C2 = C([1,21]), C3 = C1 + C2 = C([1,19]), and C∩ = C1 ∩ C2 = C([725,21]) [i] based on
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,19}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(384, 728, F3, 21) (dual of [728, 644, 22]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(397, 728, F3, 25) (dual of [728, 631, 26]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,21}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(378, 728, F3, 19) (dual of [728, 650, 20]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(35, 18, F3, 3) (dual of [18, 13, 4]-code or 18-cap in PG(4,3)), using
- 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
- construction XX applied to C1 = C([725,19]), C2 = C([1,21]), C3 = C1 + C2 = C([1,19]), and C∩ = C1 ∩ C2 = C([725,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3103, 753, F3, 25) (dual of [753, 650, 26]-code), using
(78, 78+25, 30039)-Net in Base 3 — Upper bound on s
There is no (78, 103, 30040)-net in base 3, because
- 1 times m-reduction [i] would yield (78, 102, 30040)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 638916 086267 232371 274155 566358 221123 727579 480257 > 3102 [i]