Best Known (32, 38, s)-Nets in Base 8
(32, 38, 699056)-Net over F8 — Constructive and digital
Digital (32, 38, 699056)-net over F8, using
- 81 times duplication [i] based on digital (31, 37, 699056)-net over F8, using
- net defined by OOA [i] based on linear OOA(837, 699056, F8, 6, 6) (dual of [(699056, 6), 4194299, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(837, 2097168, F8, 6) (dual of [2097168, 2097131, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(822, 2097152, F8, 4) (dual of [2097152, 2097130, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 16, F8, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
- dual of repetition code with length 16 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(837, 2097168, F8, 6) (dual of [2097168, 2097131, 7]-code), using
- net defined by OOA [i] based on linear OOA(837, 699056, F8, 6, 6) (dual of [(699056, 6), 4194299, 7]-NRT-code), using
(32, 38, 1398103)-Net in Base 8 — Constructive
(32, 38, 1398103)-net in base 8, using
- net defined by OOA [i] based on OOA(838, 1398103, S8, 6, 6), using
- OA 3-folding and stacking [i] based on OA(838, 4194309, S8, 6), using
- discarding factors based on OA(838, 4194310, S8, 6), using
- trace code [i] based on OA(6419, 2097155, S64, 6), using
- discarding parts of the base [i] based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding parts of the base [i] based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- trace code [i] based on OA(6419, 2097155, S64, 6), using
- discarding factors based on OA(838, 4194310, S8, 6), using
- OA 3-folding and stacking [i] based on OA(838, 4194309, S8, 6), using
(32, 38, 2717831)-Net over F8 — Digital
Digital (32, 38, 2717831)-net over F8, using
(32, 38, 2781737)-Net in Base 8
(32, 38, 2781737)-net in base 8, using
- net defined by OOA [i] based on OOA(838, 2781737, S8, 9, 6), using
- OOA stacking with additional row [i] based on OOA(838, 2781738, S8, 3, 6), using
- trace code [i] based on OOA(6419, 1390869, S64, 3, 6), using
- discarding parts of the base [i] based on linear OOA(12816, 1390869, F128, 3, 6) (dual of [(1390869, 3), 4172591, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12816, 1390869, F128, 6) (dual of [1390869, 1390853, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12816, 1390869, F128, 6) (dual of [1390869, 1390853, 7]-code), using
- discarding parts of the base [i] based on linear OOA(12816, 1390869, F128, 3, 6) (dual of [(1390869, 3), 4172591, 7]-NRT-code), using
- trace code [i] based on OOA(6419, 1390869, S64, 3, 6), using
- OOA stacking with additional row [i] based on OOA(838, 2781738, S8, 3, 6), using
(32, 38, large)-Net in Base 8 — Upper bound on s
There is no (32, 38, large)-net in base 8, because
- 4 times m-reduction [i] would yield (32, 34, large)-net in base 8, but