Best Known (9−4, 9, s)-Nets in Base 32
(9−4, 9, 1067)-Net over F32 — Constructive and digital
Digital (5, 9, 1067)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 33)-net over F32, using
- s-reduction based on digital (0, 0, s)-net over F32 with arbitrarily large s, using
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 1, 33)-net over F32, using
- s-reduction based on digital (0, 1, s)-net over F32 with arbitrarily large s, using
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 2, 33)-net over F32, using
- digital (1, 5, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (0, 0, 33)-net over F32, using
(9−4, 9, 2015)-Net over F32 — Digital
Digital (5, 9, 2015)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(329, 2015, F32, 4) (dual of [2015, 2006, 5]-code), using
- 986 step Varšamov–Edel lengthening with (ri) = (1, 62 times 0, 1, 922 times 0) [i] based on linear OA(327, 1027, F32, 4) (dual of [1027, 1020, 5]-code), using
- construction XX applied to C1 = C([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(327, 1023, F32, 4) (dual of [1023, 1016, 5]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(323, 1023, F32, 2) (dual of [1023, 1020, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [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([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
- 986 step Varšamov–Edel lengthening with (ri) = (1, 62 times 0, 1, 922 times 0) [i] based on linear OA(327, 1027, F32, 4) (dual of [1027, 1020, 5]-code), using
(9−4, 9, 8128)-Net in Base 32 — Constructive
(5, 9, 8128)-net in base 32, using
- net defined by OOA [i] based on OOA(329, 8128, S32, 4, 4), using
- OA 2-folding and stacking [i] based on OA(329, 16256, S32, 4), using
- discarding parts of the base [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- 1 times truncation [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- discarding parts of the base [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- OA 2-folding and stacking [i] based on OA(329, 16256, S32, 4), using
(9−4, 9, 270599)-Net in Base 32 — Upper bound on s
There is no (5, 9, 270600)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 35 184451 780501 > 329 [i]