Best Known (88−20, 88, s)-Nets in Base 3
(88−20, 88, 464)-Net over F3 — Constructive and digital
Digital (68, 88, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 22, 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
(88−20, 88, 748)-Net over F3 — Digital
Digital (68, 88, 748)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(388, 748, F3, 20) (dual of [748, 660, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(388, 764, F3, 20) (dual of [764, 676, 21]-code), using
- construction XX applied to C1 = C([346,363]), C2 = C([352,365]), C3 = C1 + C2 = C([352,363]), and C∩ = C1 ∩ C2 = C([346,365]) [i] based on
- linear OA(372, 728, F3, 18) (dual of [728, 656, 19]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {346,347,…,363}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(355, 728, F3, 14) (dual of [728, 673, 15]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {352,353,…,365}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {346,347,…,365}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(348, 728, F3, 12) (dual of [728, 680, 13]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {352,353,…,363}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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([346,363]), C2 = C([352,365]), C3 = C1 + C2 = C([352,363]), and C∩ = C1 ∩ C2 = C([346,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(388, 764, F3, 20) (dual of [764, 676, 21]-code), using
(88−20, 88, 35768)-Net in Base 3 — Upper bound on s
There is no (68, 88, 35769)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 969969 514972 579321 362681 698000 349604 865497 > 388 [i]