Best Known (29−5, 29, s)-Nets in Base 5
(29−5, 29, 39066)-Net over F5 — Constructive and digital
Digital (24, 29, 39066)-net over F5, using
- net defined by OOA [i] based on linear OOA(529, 39066, F5, 5, 5) (dual of [(39066, 5), 195301, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(529, 78133, F5, 5) (dual of [78133, 78104, 6]-code), using
- 1 times truncation [i] based on linear OA(530, 78134, F5, 6) (dual of [78134, 78104, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(522, 78125, F5, 4) (dual of [78125, 78103, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- dual of repetition code with length 9 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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
- 1 times truncation [i] based on linear OA(530, 78134, F5, 6) (dual of [78134, 78104, 7]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(529, 78133, F5, 5) (dual of [78133, 78104, 6]-code), using
(29−5, 29, 78133)-Net over F5 — Digital
Digital (24, 29, 78133)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(529, 78133, F5, 5) (dual of [78133, 78104, 6]-code), using
- 1 times truncation [i] based on linear OA(530, 78134, F5, 6) (dual of [78134, 78104, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(522, 78125, F5, 4) (dual of [78125, 78103, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- dual of repetition code with length 9 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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
- 1 times truncation [i] based on linear OA(530, 78134, F5, 6) (dual of [78134, 78104, 7]-code), using
(29−5, 29, large)-Net in Base 5 — Upper bound on s
There is no (24, 29, large)-net in base 5, because
- 3 times m-reduction [i] would yield (24, 26, large)-net in base 5, but