Best Known (64−11, 64, s)-Nets in Base 32
(64−11, 64, 1694105)-Net over F32 — Constructive and digital
Digital (53, 64, 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 (40, 51, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (8, 13, 16385)-net over F32, using
(64−11, 64, 1710617)-Net in Base 32 — Constructive
(53, 64, 1710617)-net in base 32, using
- base change [i] based on digital (29, 40, 1710617)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 32897)-net over F256, using
- net defined by OOA [i] based on linear OOA(2569, 32897, F256, 5, 5) (dual of [(32897, 5), 164476, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(2569, 32897, F256, 4, 5) (dual of [(32897, 4), 131579, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2562, 257, F256, 4, 2) (dual of [(257, 4), 1026, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;1026,256) [i]
- linear OOA(2567, 32640, F256, 4, 5) (dual of [(32640, 4), 130553, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- linear OOA(2562, 257, F256, 4, 2) (dual of [(257, 4), 1026, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(2569, 32897, F256, 4, 5) (dual of [(32897, 4), 131579, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2569, 32897, F256, 5, 5) (dual of [(32897, 5), 164476, 6]-NRT-code), using
- digital (20, 31, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- digital (4, 9, 32897)-net over F256, using
- (u, u+v)-construction [i] based on
(64−11, 64, large)-Net over F32 — Digital
Digital (53, 64, large)-net over F32, using
- t-expansion [i] based on digital (52, 64, large)-net over F32, using
- 2 times m-reduction [i] based on digital (52, 66, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3266, large, F32, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3266, large, F32, 14) (dual of [large, large−66, 15]-code), using
- 2 times m-reduction [i] based on digital (52, 66, large)-net over F32, using
(64−11, 64, large)-Net in Base 32 — Upper bound on s
There is no (53, 64, large)-net in base 32, because
- 9 times m-reduction [i] would yield (53, 55, large)-net in base 32, but