Best Known (55−27, 55, s)-Nets in Base 32
(55−27, 55, 196)-Net over F32 — Constructive and digital
Digital (28, 55, 196)-net over F32, using
- 1 times m-reduction [i] based on digital (28, 56, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (7, 35, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 21, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(55−27, 55, 288)-Net in Base 32 — Constructive
(28, 55, 288)-net in base 32, using
- 11 times m-reduction [i] based on (28, 66, 288)-net in base 32, using
- base change [i] based on (17, 55, 288)-net in base 64, using
- 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 48, 288)-net over F128, using
- 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on (17, 55, 288)-net in base 64, using
(55−27, 55, 574)-Net over F32 — Digital
Digital (28, 55, 574)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3255, 574, F32, 27) (dual of [574, 519, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3255, 1032, F32, 27) (dual of [1032, 977, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- linear OA(3253, 1024, F32, 27) (dual of [1024, 971, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3247, 1024, F32, 24) (dual of [1024, 977, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(3255, 1032, F32, 27) (dual of [1032, 977, 28]-code), using
(55−27, 55, 326736)-Net in Base 32 — Upper bound on s
There is no (28, 55, 326737)-net in base 32, because
- 1 times m-reduction [i] would yield (28, 54, 326737)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1897 189654 853911 585292 194084 719615 485549 195482 493354 389429 402398 527793 232867 821840 > 3254 [i]