Best Known (88−21, 88, s)-Nets in Base 32
(88−21, 88, 104861)-Net over F32 — Constructive and digital
Digital (67, 88, 104861)-net over F32, using
- net defined by OOA [i] based on linear OOA(3288, 104861, F32, 21, 21) (dual of [(104861, 21), 2201993, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3288, 1048611, F32, 21) (dual of [1048611, 1048523, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(3281, 1048576, F32, 21) (dual of [1048576, 1048495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(327, 35, F32, 6) (dual of [35, 28, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(327, 43, F32, 6) (dual of [43, 36, 7]-code), using
- algebraic-geometric code AG(F,36P) [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- discarding factors / shortening the dual code based on linear OA(327, 43, F32, 6) (dual of [43, 36, 7]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(3288, 1048611, F32, 21) (dual of [1048611, 1048523, 22]-code), using
(88−21, 88, 209716)-Net in Base 32 — Constructive
(67, 88, 209716)-net in base 32, using
- net defined by OOA [i] based on OOA(3288, 209716, S32, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3288, 2097161, S32, 21), using
- 1 times code embedding in larger space [i] based on OA(3287, 2097160, S32, 21), using
- discarding parts of the base [i] based on linear OA(12862, 2097160, F128, 21) (dual of [2097160, 2097098, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding parts of the base [i] based on linear OA(12862, 2097160, F128, 21) (dual of [2097160, 2097098, 22]-code), using
- 1 times code embedding in larger space [i] based on OA(3287, 2097160, S32, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3288, 2097161, S32, 21), using
(88−21, 88, 1123591)-Net over F32 — Digital
Digital (67, 88, 1123591)-net over F32, using
(88−21, 88, large)-Net in Base 32 — Upper bound on s
There is no (67, 88, large)-net in base 32, because
- 19 times m-reduction [i] would yield (67, 69, large)-net in base 32, but