Best Known (31, 31+33, s)-Nets in Base 32
(31, 31+33, 196)-Net over F32 — Constructive and digital
Digital (31, 64, 196)-net over F32, using
- 1 times m-reduction [i] based on digital (31, 65, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 24, 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, 41, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 24, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(31, 31+33, 288)-Net in Base 32 — Constructive
(31, 64, 288)-net in base 32, using
- 13 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
- base change [i] based on digital (9, 55, 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, 55, 288)-net over F128, using
(31, 31+33, 486)-Net over F32 — Digital
Digital (31, 64, 486)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3264, 486, F32, 2, 33) (dual of [(486, 2), 908, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3264, 513, F32, 2, 33) (dual of [(513, 2), 962, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3264, 1026, F32, 33) (dual of [1026, 962, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3264, 1027, F32, 33) (dual of [1027, 963, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(3263, 1024, F32, 33) (dual of [1024, 961, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3261, 1024, F32, 31) (dual of [1024, 963, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(321, 3, F32, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3264, 1027, F32, 33) (dual of [1027, 963, 34]-code), using
- OOA 2-folding [i] based on linear OA(3264, 1026, F32, 33) (dual of [1026, 962, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3264, 513, F32, 2, 33) (dual of [(513, 2), 962, 34]-NRT-code), using
(31, 31+33, 185219)-Net in Base 32 — Upper bound on s
There is no (31, 64, 185220)-net in base 32, because
- 1 times m-reduction [i] would yield (31, 63, 185220)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 66753 541121 757024 022441 757644 370461 255140 637329 506261 178951 742478 743646 385198 989732 511255 966088 > 3263 [i]