Best Known (15−4, 15, s)-Nets in Base 32
(15−4, 15, 524323)-Net over F32 — Constructive and digital
Digital (11, 15, 524323)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 33)-net over F32, using
- digital (9, 13, 524290)-net over F32, using
- net defined by OOA [i] based on linear OOA(3213, 524290, F32, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(3213, 1048580, F32, 4) (dual of [1048580, 1048567, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(3213, 1048576, F32, 4) (dual of [1048576, 1048563, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(329, 1048576, F32, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 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 X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(3213, 1048580, F32, 4) (dual of [1048580, 1048567, 5]-code), using
- net defined by OOA [i] based on linear OOA(3213, 524290, F32, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
(15−4, 15, 1048578)-Net in Base 32 — Constructive
(11, 15, 1048578)-net in base 32, using
- net defined by OOA [i] based on OOA(3215, 1048578, S32, 4, 4), using
- OA 2-folding and stacking [i] based on OA(3215, 2097156, S32, 4), using
- 1 times code embedding in larger space [i] based on OA(3214, 2097155, S32, 4), using
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- 1 times code embedding in larger space [i] based on OA(3214, 2097155, S32, 4), using
- OA 2-folding and stacking [i] based on OA(3215, 2097156, S32, 4), using
(15−4, 15, 1966855)-Net over F32 — Digital
Digital (11, 15, 1966855)-net over F32, using
(15−4, 15, 2097155)-Net in Base 32
(11, 15, 2097155)-net in base 32, using
- 321 times duplication [i] based on (10, 14, 2097155)-net in base 32, using
- base change [i] based on digital (6, 10, 2097155)-net over F128, using
- net defined by OOA [i] based on linear OOA(12810, 2097155, F128, 4, 4) (dual of [(2097155, 4), 8388610, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(12810, 2097155, F128, 3, 4) (dual of [(2097155, 3), 6291455, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- appending kth column [i] based on linear OOA(12810, 2097155, F128, 3, 4) (dual of [(2097155, 3), 6291455, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(12810, 2097155, F128, 4, 4) (dual of [(2097155, 4), 8388610, 5]-NRT-code), using
- base change [i] based on digital (6, 10, 2097155)-net over F128, using
(15−4, 15, large)-Net in Base 32 — Upper bound on s
There is no (11, 15, large)-net in base 32, because
- 2 times m-reduction [i] would yield (11, 13, large)-net in base 32, but