Best Known (49, 49+10, s)-Nets in Base 32
(49, 49+10, 1694105)-Net over F32 — Constructive and digital
Digital (49, 59, 1694105)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 16385)-net over F32, using
- net defined by OOA [i] based on linear OOA(3213, 16385, F32, 5, 5) (dual of [(16385, 5), 81912, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3213, 32771, F32, 5) (dual of [32771, 32758, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(3213, 32768, F32, 5) (dual of [32768, 32755, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(3210, 32768, F32, 4) (dual of [32768, 32758, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(3213, 32771, F32, 5) (dual of [32771, 32758, 6]-code), using
- net defined by OOA [i] based on linear OOA(3213, 16385, F32, 5, 5) (dual of [(16385, 5), 81912, 6]-NRT-code), using
- digital (36, 46, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (8, 13, 16385)-net over F32, using
(49, 49+10, 1710489)-Net in Base 32 — Constructive
(49, 59, 1710489)-net in base 32, using
- (u, u+v)-construction [i] based on
- (9, 14, 32769)-net in base 32, using
- base change [i] based on (5, 10, 32769)-net in base 128, using
- net defined by OOA [i] based on OOA(12810, 32769, S128, 5, 5), using
- appending kth column [i] based on OOA(12810, 32769, S128, 4, 5), using
- (u, u+v)-construction [i] based on
- linear OOA(1282, 129, F128, 4, 2) (dual of [(129, 4), 514, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;514,128) [i]
- OOA(1288, 32640, S128, 4, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(1288, 65281, S128, 5), using
- discarding parts of the base [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(1288, 65281, S128, 5), using
- linear OOA(1282, 129, F128, 4, 2) (dual of [(129, 4), 514, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on OOA(12810, 32769, S128, 4, 5), using
- net defined by OOA [i] based on OOA(12810, 32769, S128, 5, 5), using
- base change [i] based on (5, 10, 32769)-net in base 128, using
- (35, 45, 1677720)-net in base 32, using
- net defined by OOA [i] based on OOA(3245, 1677720, S32, 10, 10), using
- OA 5-folding and stacking [i] based on OA(3245, 8388600, S32, 10), using
- discarding factors based on OA(3245, large, S32, 10), using
- discarding parts of the base [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding parts of the base [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- discarding factors based on OA(3245, large, S32, 10), using
- OA 5-folding and stacking [i] based on OA(3245, 8388600, S32, 10), using
- net defined by OOA [i] based on OOA(3245, 1677720, S32, 10, 10), using
- (9, 14, 32769)-net in base 32, using
(49, 49+10, large)-Net over F32 — Digital
Digital (49, 59, large)-net over F32, using
- t-expansion [i] based on digital (48, 59, large)-net over F32, using
- 2 times m-reduction [i] based on digital (48, 61, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- 2 times m-reduction [i] based on digital (48, 61, large)-net over F32, using
(49, 49+10, large)-Net in Base 32 — Upper bound on s
There is no (49, 59, large)-net in base 32, because
- 8 times m-reduction [i] would yield (49, 51, large)-net in base 32, but