MinT - Best Known (139, 139+16, s)-Nets in Base 8

Best Known (139, 139+16, s)-Nets in Base 8

(139, 139+16, 2099200)-Net over F₈ — Constructive and digital

Digital (139, 155, 2099200)-net over F₈, using

8¹ times duplication [i] based on digital (138, 154, 2099200)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (24, 32, 2050)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8³², 2050, F₈, 8, 8) (dual of [(2050, 8), 16368, 9]-NRT-code), using
      - OA 4-folding and stacking [i] based on linear OA(8³², 8200, F₈, 8) (dual of [8200, 8168, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(8³², 8202, F₈, 8) (dual of [8202, 8170, 9]-code), using
        trace code [i] based on linear OA(64¹⁶, 4101, F₆₄, 8) (dual of [4101, 4085, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(5) [i] based on
        linear OA(64¹⁵, 4096, F₆₄, 8) (dual of [4096, 4081, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(64¹¹, 4096, F₆₄, 6) (dual of [4096, 4085, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
  2. digital (106, 122, 2097150)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8¹²², 2097150, F₈, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹²², 8388601, F₈, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(8¹²², 8388602, F₈, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
        trace code [i] based on linear OOA(64⁶¹, 4194301, F₆₄, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶¹, 8388602, F₆₄, 16) (dual of [8388602, 8388541, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶¹, large, F₆₄, 16) (dual of [large, large−61, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(139, 139+16, large)-Net over F₈ — Digital

Digital (139, 155, large)-net over F₈, using

t-expansion [i] based on digital (138, 155, large)-net over F₈, using
- 6 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(139, 139+16, large)-Net in Base 8 — Upper bound on s

There is no (139, 155, large)-net in base 8, because

14 times m-reduction [i] would yield (139, 141, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (139, 139+16, s)-Nets in Base 8

(139, 139+16, 2099200)-Net over F8 — Constructive and digital

(139, 139+16, large)-Net over F8 — Digital

(139, 139+16, large)-Net in Base 8 — Upper bound on s

(139, 139+16, 2099200)-Net over F₈ — Constructive and digital

(139, 139+16, large)-Net over F₈ — Digital