Best Known (73−21, 73, s)-Nets in Base 3
(73−21, 73, 156)-Net over F3 — Constructive and digital
Digital (52, 73, 156)-net over F3, using
- 31 times duplication [i] based on digital (51, 72, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 24, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 24, 52)-net over F27, using
(73−21, 73, 240)-Net over F3 — Digital
Digital (52, 73, 240)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(373, 240, F3, 21) (dual of [240, 167, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 259, F3, 21) (dual of [259, 186, 22]-code), using
- construction XX applied to C1 = C([239,16]), C2 = C([0,18]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([239,18]) [i] based on
- linear OA(366, 242, F3, 20) (dual of [242, 176, 21]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,16}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(361, 242, F3, 19) (dual of [242, 181, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(371, 242, F3, 22) (dual of [242, 171, 23]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,18}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(31, 6, F3, 1) (dual of [6, 5, 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 (see above)
- construction XX applied to C1 = C([239,16]), C2 = C([0,18]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([239,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(373, 259, F3, 21) (dual of [259, 186, 22]-code), using
(73−21, 73, 6159)-Net in Base 3 — Upper bound on s
There is no (52, 73, 6160)-net in base 3, because
- 1 times m-reduction [i] would yield (52, 72, 6160)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 22544 068283 891676 285559 766486 061921 > 372 [i]