Best Known (59, 59+21, s)-Nets in Base 32
(59, 59+21, 3482)-Net over F32 — Constructive and digital
Digital (59, 80, 3482)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 19, 205)-net over F32, using
- net defined by OOA [i] based on linear OOA(3219, 205, F32, 10, 10) (dual of [(205, 10), 2031, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3219, 1025, F32, 10) (dual of [1025, 1006, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 1026, F32, 10) (dual of [1026, 1007, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(3219, 1024, F32, 10) (dual of [1024, 1005, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3217, 1024, F32, 9) (dual of [1024, 1007, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(3219, 1026, F32, 10) (dual of [1026, 1007, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3219, 1025, F32, 10) (dual of [1025, 1006, 11]-code), using
- net defined by OOA [i] based on linear OOA(3219, 205, F32, 10, 10) (dual of [(205, 10), 2031, 11]-NRT-code), using
- digital (40, 61, 3277)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 3277, F32, 21, 21) (dual of [(3277, 21), 68756, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3261, 32771, F32, 21) (dual of [32771, 32710, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(3261, 32768, F32, 21) (dual of [32768, 32707, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(3258, 32768, F32, 20) (dual of [32768, 32710, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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 (see above)
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(3261, 32771, F32, 21) (dual of [32771, 32710, 22]-code), using
- net defined by OOA [i] based on linear OOA(3261, 3277, F32, 21, 21) (dual of [(3277, 21), 68756, 22]-NRT-code), using
- digital (9, 19, 205)-net over F32, using
(59, 59+21, 26216)-Net in Base 32 — Constructive
(59, 80, 26216)-net in base 32, using
- 322 times duplication [i] based on (57, 78, 26216)-net in base 32, using
- base change [i] based on digital (44, 65, 26216)-net over F64, using
- net defined by OOA [i] based on linear OOA(6465, 26216, F64, 21, 21) (dual of [(26216, 21), 550471, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6465, 262161, F64, 21) (dual of [262161, 262096, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, 262163, F64, 21) (dual of [262163, 262098, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(644, 19, F64, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(6465, 262163, F64, 21) (dual of [262163, 262098, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6465, 262161, F64, 21) (dual of [262161, 262096, 22]-code), using
- net defined by OOA [i] based on linear OOA(6465, 26216, F64, 21, 21) (dual of [(26216, 21), 550471, 22]-NRT-code), using
- base change [i] based on digital (44, 65, 26216)-net over F64, using
(59, 59+21, 280905)-Net over F32 — Digital
Digital (59, 80, 280905)-net over F32, using
(59, 59+21, large)-Net in Base 32 — Upper bound on s
There is no (59, 80, large)-net in base 32, because
- 19 times m-reduction [i] would yield (59, 61, large)-net in base 32, but