MinT - Best Known (23, 33, s)-Nets in Base 8

Best Known (23, 33, s)-Nets in Base 8

(23, 33, 820)-Net over F₈ — Constructive and digital

Digital (23, 33, 820)-net over F₈, using

net defined by OOA [i] based on linear OOA(8³³, 820, F₈, 10, 10) (dual of [(820, 10), 8167, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8³³, 4100, F₈, 10) (dual of [4100, 4067, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
    1. linear OA(8³³, 4096, F₈, 10) (dual of [4096, 4063, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(8²⁹, 4096, F₈, 9) (dual of [4096, 4067, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(23, 33, 2199)-Net over F₈ — Digital

Digital (23, 33, 2199)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³³, 2199, F₈, 10) (dual of [2199, 2166, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8³³, 4096, F₈, 10) (dual of [4096, 4063, 11]-code), using
  - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(23, 33, 339725)-Net in Base 8 — Upper bound on s

There is no (23, 33, 339726)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 633826 264577 954508 428254 490633 > 8³³ [i]