Best Known (48−24, 48, s)-Nets in Base 32
(48−24, 48, 174)-Net over F32 — Constructive and digital
Digital (24, 48, 174)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- digital (7, 31, 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 (5, 17, 76)-net over F32, using
(48−24, 48, 288)-Net in Base 32 — Constructive
(24, 48, 288)-net in base 32, using
- t-expansion [i] based on (23, 48, 288)-net in base 32, using
- 1 times m-reduction [i] based on (23, 49, 288)-net in base 32, using
- base change [i] based on digital (9, 35, 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, 35, 288)-net over F128, using
- 1 times m-reduction [i] based on (23, 49, 288)-net in base 32, using
(48−24, 48, 515)-Net over F32 — Digital
Digital (24, 48, 515)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3248, 515, F32, 2, 24) (dual of [(515, 2), 982, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3248, 1030, F32, 24) (dual of [1030, 982, 25]-code), using
- construction XX applied to C1 = C([1021,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1021,21]) [i] based on
- linear OA(3245, 1023, F32, 23) (dual of [1023, 978, 24]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(3243, 1023, F32, 22) (dual of [1023, 980, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3247, 1023, F32, 24) (dual of [1023, 976, 25]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,21}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3241, 1023, F32, 21) (dual of [1023, 982, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- 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
- construction XX applied to C1 = C([1021,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1021,21]) [i] based on
- OOA 2-folding [i] based on linear OA(3248, 1030, F32, 24) (dual of [1030, 982, 25]-code), using
(48−24, 48, 178889)-Net in Base 32 — Upper bound on s
There is no (24, 48, 178890)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1 766882 352647 639353 240906 893751 023589 819395 725849 709498 797772 761422 056048 > 3248 [i]