Basicu matrix system is currently the 2nd fastest function that can be computed in the Googology with the notation-type expression. I like to thanks to PsiCubed2 to improve my encoding!

There are 15 symbols being used in BMS:

  • ( and ) brackets (Groups of 2 symbols)
  • [ and ] brackets (Groups of 2 symbols)
  • A comma
  • Integers (Groups of 10 symbols)

Today, I can encode to few symbols. I start encoding to 12-symbol system.


First, you can remove the rightmost leading 0s in any () bracket before you start to encode the expression. And do the reduction in every () brackets!

To reduce more, remove the (0) bracket.


12-symbol system

To encode to 12-symbol system, remove the (, [, and ] then replace the ) with a period.

11-symbol system

To encode to 11-symbol system from 12-symbol system, just replace a period with 2 commas.

10-symbol system

To encode to 10-symbol system from 11-symbol system, convert all decimal numbers to base 9 numbers. To go with smaller amount of symbols, convert decimal to the lower base system.

Converting to Unary doesn't work, but I can encode to 2 symbols.

2-symbol system

To encode to 2-symbol system from 11-symbol system, each integer you see are increased by 1, then convert the number to amount of 0s.

1-symbol system

Finally, to encode to 1-symbol system from 2-symbol system, replace a comma with 1, convert the entire encoded text from binary to decimal, then the result will be the converted number amount of 0s.


You can decode easily by reversing the encoding process, but decoding 12-symbol to 15-symbol is hard.


  • \((0,0,0)(1,1,0)(2,2,1)(3,3,1)[4]\) (Start) =>
  • \((1,1)(2,2,1)(3,3,1)[4]\) (Reduction) =>
  • \(1,1.2,2,1.3,3,1.4\) (12-symbol system) =>
  • \(1,1,,2,2,1,,3,3,1,,4\) (11-symbol system) =>
  • \(2,2,,3,3,2,,4,4,2,,5\) (Increase every number by 1) =>
  • \(00,00,,000,000,00,,0000,0000,00,,00000\) (Convert to unary, replacing the number to amount of 0s, 2-symbol system) =>
  • \(00100110001000100110000100001001100000\) (Replace every commas with 1) =>
  • \(40946385504\) (Convert binary to decimal) =>
  • \(0^{40946385504}\) (Convert to unary, replacing it with the number to amount of 0s, 1-symbol system, end)