MinT - Best Known (67−14, 67, s)-Nets in Base 5

Best Known (67−14, 67, s)-Nets in Base 5

(67−14, 67, 2233)-Net over F₅ — Constructive and digital

Digital (53, 67, 2233)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁶⁷, 2233, F₅, 14, 14) (dual of [(2233, 14), 31195, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(5⁶⁷, 15631, F₅, 14) (dual of [15631, 15564, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
    1. linear OA(5⁶⁷, 15625, F₅, 14) (dual of [15625, 15558, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(5⁶¹, 15625, F₅, 13) (dual of [15625, 15564, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(5⁰, 6, F₅, 0) (dual of [6, 6, 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
        OA with only one run / dual of code without redundancy [i]

(67−14, 67, 9231)-Net over F₅ — Digital

Digital (53, 67, 9231)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁶⁷, 9231, F₅, 14) (dual of [9231, 9164, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁶⁷, 15625, F₅, 14) (dual of [15625, 15558, 15]-code), using
  - an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(67−14, 67, 4139993)-Net in Base 5 — Upper bound on s

There is no (53, 67, 4139994)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 67762 726634 459128 852926 255859 615749 609931 050121 > 5⁶⁷ [i]