Best Known (39−6, 39, s)-Nets in Base 5
(39−6, 39, 651045)-Net over F5 — Constructive and digital
Digital (33, 39, 651045)-net over F5, using
- 51 times duplication [i] based on digital (32, 38, 651045)-net over F5, using
- net defined by OOA [i] based on linear OOA(538, 651045, F5, 6, 6) (dual of [(651045, 6), 3906232, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(538, 1953135, F5, 6) (dual of [1953135, 1953097, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 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 X applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(538, 1953135, F5, 6) (dual of [1953135, 1953097, 7]-code), using
- net defined by OOA [i] based on linear OOA(538, 651045, F5, 6, 6) (dual of [(651045, 6), 3906232, 7]-NRT-code), using
(39−6, 39, 1953138)-Net over F5 — Digital
Digital (33, 39, 1953138)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(539, 1953138, F5, 6) (dual of [1953138, 1953099, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(538, 1953136, F5, 6) (dual of [1953136, 1953098, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(510, 11, F5, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,5)), using
- dual of repetition code with length 11 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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 Ce(5) ⊂ Ce(3) [i] based on
- linear OA(538, 1953137, F5, 5) (dual of [1953137, 1953099, 6]-code), using Gilbert–Varšamov bound and bm = 538 > Vbs−1(k−1) = 155 223528 567291 336789 158465 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
- linear OA(538, 1953136, F5, 6) (dual of [1953136, 1953098, 7]-code), using
- construction X with Varšamov bound [i] based on
(39−6, 39, large)-Net in Base 5 — Upper bound on s
There is no (33, 39, large)-net in base 5, because
- 4 times m-reduction [i] would yield (33, 35, large)-net in base 5, but