Best Known (18, 18+18, s)-Nets in Base 32
(18, 18+18, 142)-Net over F32 — Constructive and digital
Digital (18, 36, 142)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 66)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (0, 9, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 4, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 23, 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 (4, 13, 66)-net over F32, using
(18, 18+18, 261)-Net in Base 32 — Constructive
(18, 36, 261)-net in base 32, using
- base change [i] based on (12, 30, 261)-net in base 64, using
- 2 times m-reduction [i] based on (12, 32, 261)-net in base 64, using
- base change [i] based on digital (4, 24, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- base change [i] based on digital (4, 24, 261)-net over F256, using
- 2 times m-reduction [i] based on (12, 32, 261)-net in base 64, using
(18, 18+18, 515)-Net over F32 — Digital
Digital (18, 36, 515)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3236, 515, F32, 2, 18) (dual of [(515, 2), 994, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3236, 1030, F32, 18) (dual of [1030, 994, 19]-code), using
- construction XX applied to C1 = C([1021,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([1021,15]) [i] based on
- linear OA(3233, 1023, F32, 17) (dual of [1023, 990, 18]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,14}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3231, 1023, F32, 16) (dual of [1023, 992, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3235, 1023, F32, 18) (dual of [1023, 988, 19]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,15}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3229, 1023, F32, 15) (dual of [1023, 994, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [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([1021,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([1021,15]) [i] based on
- OOA 2-folding [i] based on linear OA(3236, 1030, F32, 18) (dual of [1030, 994, 19]-code), using
(18, 18+18, 140273)-Net in Base 32 — Upper bound on s
There is no (18, 36, 140274)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1 532520 837339 981138 110217 549223 730096 918313 847469 055782 > 3236 [i]