Best Known (52−27, 52, s)-Nets in Base 32
(52−27, 52, 174)-Net over F32 — Constructive and digital
Digital (25, 52, 174)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 18, 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, 34, 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, 18, 76)-net over F32, using
(52−27, 52, 288)-Net in Base 32 — Constructive
(25, 52, 288)-net in base 32, using
- 4 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
- base change [i] based on digital (9, 40, 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, 40, 288)-net over F128, using
(52−27, 52, 365)-Net over F32 — Digital
Digital (25, 52, 365)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3252, 365, F32, 27) (dual of [365, 313, 28]-code), using
- 19 step Varšamov–Edel lengthening with (ri) = (1, 18 times 0) [i] based on linear OA(3251, 345, F32, 27) (dual of [345, 294, 28]-code), using
- construction XX applied to C1 = C([340,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([340,25]) [i] based on
- linear OA(3249, 341, F32, 26) (dual of [341, 292, 27]-code), using the BCH-code C(I) with length 341 | 322−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3249, 341, F32, 26) (dual of [341, 292, 27]-code), using the expurgated narrow-sense BCH-code C(I) with length 341 | 322−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3251, 341, F32, 27) (dual of [341, 290, 28]-code), using the BCH-code C(I) with length 341 | 322−1, defining interval I = {−1,0,…,25}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3247, 341, F32, 25) (dual of [341, 294, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 341 | 322−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [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([340,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([340,25]) [i] based on
- 19 step Varšamov–Edel lengthening with (ri) = (1, 18 times 0) [i] based on linear OA(3251, 345, F32, 27) (dual of [345, 294, 28]-code), using
(52−27, 52, 146840)-Net in Base 32 — Upper bound on s
There is no (25, 52, 146841)-net in base 32, because
- 1 times m-reduction [i] would yield (25, 51, 146841)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 57900 704124 435695 514956 836918 997658 651383 936261 698758 433669 435482 780402 828912 > 3251 [i]