Best Known (36−9, 36, s)-Nets in Base 5
(36−9, 36, 782)-Net over F5 — Constructive and digital
Digital (27, 36, 782)-net over F5, using
- net defined by OOA [i] based on linear OOA(536, 782, F5, 9, 9) (dual of [(782, 9), 7002, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(536, 3129, F5, 9) (dual of [3129, 3093, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(536, 3130, F5, 9) (dual of [3130, 3094, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(536, 3130, F5, 9) (dual of [3130, 3094, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(536, 3129, F5, 9) (dual of [3129, 3093, 10]-code), using
(36−9, 36, 2636)-Net over F5 — Digital
Digital (27, 36, 2636)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(536, 2636, F5, 9) (dual of [2636, 2600, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using
(36−9, 36, 722734)-Net in Base 5 — Upper bound on s
There is no (27, 36, 722735)-net in base 5, because
- 1 times m-reduction [i] would yield (27, 35, 722735)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2 910389 266974 848174 562801 > 535 [i]