Best Known (55−29, 55, s)-Nets in Base 32
(55−29, 55, 174)-Net over F32 — Constructive and digital
Digital (26, 55, 174)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 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, 36, 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, 19, 76)-net over F32, using
(55−29, 55, 288)-Net in Base 32 — Constructive
(26, 55, 288)-net in base 32, using
- t-expansion [i] based on (25, 55, 288)-net in base 32, using
- 1 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
- 1 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
(55−29, 55, 345)-Net over F32 — Digital
Digital (26, 55, 345)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3255, 345, F32, 29) (dual of [345, 290, 30]-code), using
- construction XX applied to C1 = C([340,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([340,27]) [i] based on
- linear OA(3253, 341, F32, 28) (dual of [341, 288, 29]-code), using the BCH-code C(I) with length 341 | 322−1, defining interval I = {−1,0,…,26}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3253, 341, F32, 28) (dual of [341, 288, 29]-code), using the expurgated narrow-sense BCH-code C(I) with length 341 | 322−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3255, 341, F32, 29) (dual of [341, 286, 30]-code), using the BCH-code C(I) with length 341 | 322−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(3251, 341, F32, 27) (dual of [341, 290, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 341 | 322−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [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([340,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([340,27]) [i] based on
(55−29, 55, 124637)-Net in Base 32 — Upper bound on s
There is no (26, 55, 124638)-net in base 32, because
- 1 times m-reduction [i] would yield (26, 54, 124638)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1897 147352 158539 802743 903862 255271 805480 964470 841754 035252 994498 112639 460029 290952 > 3254 [i]