Best Known (67, 67+23, s)-Nets in Base 32
(67, 67+23, 95325)-Net over F32 — Constructive and digital
Digital (67, 90, 95325)-net over F32, using
- 321 times duplication [i] based on digital (66, 89, 95325)-net over F32, using
- net defined by OOA [i] based on linear OOA(3289, 95325, F32, 23, 23) (dual of [(95325, 23), 2192386, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3289, 1048576, F32, 23) (dual of [1048576, 1048487, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 11-folding and stacking with additional row [i] based on linear OA(3289, 1048576, F32, 23) (dual of [1048576, 1048487, 24]-code), using
- net defined by OOA [i] based on linear OOA(3289, 95325, F32, 23, 23) (dual of [(95325, 23), 2192386, 24]-NRT-code), using
(67, 67+23, 670028)-Net over F32 — Digital
Digital (67, 90, 670028)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3290, 670028, F32, 23) (dual of [670028, 669938, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3290, 1048586, F32, 23) (dual of [1048586, 1048496, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(3289, 1048577, F32, 23) (dual of [1048577, 1048488, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(3281, 1048577, F32, 21) (dual of [1048577, 1048496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3290, 1048586, F32, 23) (dual of [1048586, 1048496, 24]-code), using
(67, 67+23, large)-Net in Base 32 — Upper bound on s
There is no (67, 90, large)-net in base 32, because
- 21 times m-reduction [i] would yield (67, 69, large)-net in base 32, but