MinT - Best Known (113−10, 113, s)-Nets in Base 8

Best Known (113−10, 113, s)-Nets in Base 8

(113−10, 113, 6711575)-Net over F₈ — Constructive and digital

Digital (103, 113, 6711575)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 6, 695)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁶, 695, F₈, 3, 3) (dual of [(695, 3), 2079, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(8⁶, 695, F₈, 2, 3) (dual of [(695, 2), 1384, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁶, 695, F₈, 3) (dual of [695, 689, 4]-code or 695-cap in PG(5,8)), using
        caps completed using a computer search [i]
2. digital (28, 33, 3355440)-net over F₈, using
  - s-reduction based on digital (28, 33, 4194301)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8³³, 4194301, F₈, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(8³³, large, F₈, 5) (dual of [large, large−33, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
3. digital (64, 74, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (27, 37, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(113−10, 113, large)-Net over F₈ — Digital

Digital (103, 113, large)-net over F₈, using

t-expansion [i] based on digital (100, 113, large)-net over F₈, using
- 4 times m-reduction [i] based on digital (100, 117, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁷, large, F₈, 17) (dual of [large, large−117, 18]-code), using
    - 4 times code embedding in larger space [i] based on linear OA(8¹¹³, large, F₈, 17) (dual of [large, large−113, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(113−10, 113, large)-Net in Base 8 — Upper bound on s

There is no (103, 113, large)-net in base 8, because

8 times m-reduction [i] would yield (103, 105, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (113−10, 113, s)-Nets in Base 8

(113−10, 113, 6711575)-Net over F8 — Constructive and digital

(113−10, 113, large)-Net over F8 — Digital

(113−10, 113, large)-Net in Base 8 — Upper bound on s

(113−10, 113, 6711575)-Net over F₈ — Constructive and digital

(113−10, 113, large)-Net over F₈ — Digital