Best Known (43, 43+15, s)-Nets in Base 32
(43, 43+15, 149797)-Net over F32 — Constructive and digital
Digital (43, 58, 149797)-net over F32, using
- 321 times duplication [i] based on digital (42, 57, 149797)-net over F32, using
- net defined by OOA [i] based on linear OOA(3257, 149797, F32, 15, 15) (dual of [(149797, 15), 2246898, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3257, 1048580, F32, 15) (dual of [1048580, 1048523, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(3257, 1048576, F32, 15) (dual of [1048576, 1048519, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3253, 1048576, F32, 14) (dual of [1048576, 1048523, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(3257, 1048580, F32, 15) (dual of [1048580, 1048523, 16]-code), using
- net defined by OOA [i] based on linear OOA(3257, 149797, F32, 15, 15) (dual of [(149797, 15), 2246898, 16]-NRT-code), using
(43, 43+15, 727017)-Net over F32 — Digital
Digital (43, 58, 727017)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3258, 727017, F32, 15) (dual of [727017, 726959, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3258, 1048586, F32, 15) (dual of [1048586, 1048528, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3257, 1048577, F32, 15) (dual of [1048577, 1048520, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(3249, 1048577, F32, 13) (dual of [1048577, 1048528, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3258, 1048586, F32, 15) (dual of [1048586, 1048528, 16]-code), using
(43, 43+15, large)-Net in Base 32 — Upper bound on s
There is no (43, 58, large)-net in base 32, because
- 13 times m-reduction [i] would yield (43, 45, large)-net in base 32, but