Best Known (31, 31+32, s)-Nets in Base 32
(31, 31+32, 196)-Net over F32 — Constructive and digital
Digital (31, 63, 196)-net over F32, using
- 2 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+32, 288)-Net in Base 32 — Constructive
(31, 63, 288)-net in base 32, using
- 14 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+32, 513)-Net over F32 — Digital
Digital (31, 63, 513)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3263, 513, F32, 2, 32) (dual of [(513, 2), 963, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3263, 1026, F32, 32) (dual of [1026, 963, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3263, 1027, F32, 32) (dual of [1027, 964, 33]-code), using
- construction XX applied to C1 = C([1022,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([1022,30]) [i] based on
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3263, 1023, F32, 32) (dual of [1023, 960, 33]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3259, 1023, F32, 30) (dual of [1023, 964, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([1022,30]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3263, 1027, F32, 32) (dual of [1027, 964, 33]-code), using
- OOA 2-folding [i] based on linear OA(3263, 1026, F32, 32) (dual of [1026, 963, 33]-code), using
(31, 31+32, 185219)-Net in Base 32 — Upper bound on s
There is no (31, 63, 185220)-net in base 32, because
- 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]