Best Known (35, 53, s)-Nets in Base 49
(35, 53, 13072)-Net over F49 — Constructive and digital
Digital (35, 53, 13072)-net over F49, using
- 491 times duplication [i] based on digital (34, 52, 13072)-net over F49, using
- net defined by OOA [i] based on linear OOA(4952, 13072, F49, 18, 18) (dual of [(13072, 18), 235244, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4952, 117648, F49, 18) (dual of [117648, 117596, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4952, 117648, F49, 18) (dual of [117648, 117596, 19]-code), using
- net defined by OOA [i] based on linear OOA(4952, 13072, F49, 18, 18) (dual of [(13072, 18), 235244, 19]-NRT-code), using
(35, 53, 58828)-Net over F49 — Digital
Digital (35, 53, 58828)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4953, 58828, F49, 2, 18) (dual of [(58828, 2), 117603, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4953, 117656, F49, 18) (dual of [117656, 117603, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4946, 117649, F49, 16) (dual of [117649, 117603, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(4953, 117656, F49, 18) (dual of [117656, 117603, 19]-code), using
(35, 53, large)-Net in Base 49 — Upper bound on s
There is no (35, 53, large)-net in base 49, because
- 16 times m-reduction [i] would yield (35, 37, large)-net in base 49, but