Best Known (53, 53+20, s)-Nets in Base 16
(53, 53+20, 6554)-Net over F16 — Constructive and digital
Digital (53, 73, 6554)-net over F16, using
- net defined by OOA [i] based on linear OOA(1673, 6554, F16, 20, 20) (dual of [(6554, 20), 131007, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1673, 65540, F16, 20) (dual of [65540, 65467, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(1673, 65536, F16, 20) (dual of [65536, 65463, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1669, 65536, F16, 19) (dual of [65536, 65467, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- OA 10-folding and stacking [i] based on linear OA(1673, 65540, F16, 20) (dual of [65540, 65467, 21]-code), using
(53, 53+20, 32991)-Net over F16 — Digital
Digital (53, 73, 32991)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1673, 32991, F16, 20) (dual of [32991, 32918, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1673, 65536, F16, 20) (dual of [65536, 65463, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(1673, 65536, F16, 20) (dual of [65536, 65463, 21]-code), using
(53, 53+20, large)-Net in Base 16 — Upper bound on s
There is no (53, 73, large)-net in base 16, because
- 18 times m-reduction [i] would yield (53, 55, large)-net in base 16, but