Best Known (9, 9+7, s)-Nets in Base 32
(9, 9+7, 1056)-Net over F32 — Constructive and digital
Digital (9, 16, 1056)-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, 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, 1, 33)-net over F32 (see above)
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 2, 33)-net over F32, using
- digital (0, 3, 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, 7, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 0, 33)-net over F32, using
(9, 9+7, 1222)-Net over F32 — Digital
Digital (9, 16, 1222)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3216, 1222, F32, 7) (dual of [1222, 1206, 8]-code), using
- 192 step Varšamov–Edel lengthening with (ri) = (2, 21 times 0, 1, 169 times 0) [i] based on linear OA(3213, 1027, F32, 7) (dual of [1027, 1014, 8]-code), using
- construction XX applied to C1 = C([1022,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1022,5]) [i] based on
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(329, 1023, F32, 5) (dual of [1023, 1014, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([1022,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1022,5]) [i] based on
- 192 step Varšamov–Edel lengthening with (ri) = (2, 21 times 0, 1, 169 times 0) [i] based on linear OA(3213, 1027, F32, 7) (dual of [1027, 1014, 8]-code), using
(9, 9+7, 1365)-Net in Base 32 — Constructive
(9, 16, 1365)-net in base 32, using
- net defined by OOA [i] based on OOA(3216, 1365, S32, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(3216, 4096, S32, 7), using
- discarding factors based on OA(3216, 4098, S32, 7), using
- discarding parts of the base [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- discarding factors based on OA(3216, 4098, S32, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(3216, 4096, S32, 7), using
(9, 9+7, 1966852)-Net in Base 32 — Upper bound on s
There is no (9, 16, 1966853)-net in base 32, because
- 1 times m-reduction [i] would yield (9, 15, 1966853)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 37778 983225 902758 745788 > 3215 [i]