Best Known (27−7, 27, s)-Nets in Base 5
(27−7, 27, 1043)-Net over F5 — Constructive and digital
Digital (20, 27, 1043)-net over F5, using
- 51 times duplication [i] based on digital (19, 26, 1043)-net over F5, using
- net defined by OOA [i] based on linear OOA(526, 1043, F5, 7, 7) (dual of [(1043, 7), 7275, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
- net defined by OOA [i] based on linear OOA(526, 1043, F5, 7, 7) (dual of [(1043, 7), 7275, 8]-NRT-code), using
(27−7, 27, 2805)-Net over F5 — Digital
Digital (20, 27, 2805)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(527, 2805, F5, 7) (dual of [2805, 2778, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(527, 3132, F5, 7) (dual of [3132, 3105, 8]-code), using
- construction X4 applied to C([0,6]) ⊂ C([1,5]) [i] based on
- linear OA(526, 3124, F5, 7) (dual of [3124, 3098, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(520, 3124, F5, 5) (dual of [3124, 3104, 6]-code), using the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(57, 8, F5, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,5)), using
- dual of repetition code with length 8 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,6]) ⊂ C([1,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(527, 3132, F5, 7) (dual of [3132, 3105, 8]-code), using
(27−7, 27, 518874)-Net in Base 5 — Upper bound on s
There is no (20, 27, 518875)-net in base 5, because
- 1 times m-reduction [i] would yield (20, 26, 518875)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1 490120 388459 790501 > 526 [i]