Best Known (20, 20+6, s)-Nets in Base 7
(20, 20+6, 5604)-Net over F7 — Constructive and digital
Digital (20, 26, 5604)-net over F7, using
- net defined by OOA [i] based on linear OOA(726, 5604, F7, 6, 6) (dual of [(5604, 6), 33598, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(726, 16812, F7, 6) (dual of [16812, 16786, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(726, 16807, F7, 6) (dual of [16807, 16781, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(721, 16807, F7, 5) (dual of [16807, 16786, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(726, 16812, F7, 6) (dual of [16812, 16786, 7]-code), using
(20, 20+6, 16812)-Net over F7 — Digital
Digital (20, 26, 16812)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(726, 16812, F7, 6) (dual of [16812, 16786, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(726, 16807, F7, 6) (dual of [16807, 16781, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(721, 16807, F7, 5) (dual of [16807, 16786, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
(20, 20+6, 6388742)-Net in Base 7 — Upper bound on s
There is no (20, 26, 6388743)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 9387 480783 983248 929583 > 726 [i]