Free Lie algebra

In telecommunications, 6b/8b is a line code that expands 6-bit codes to 8-bit symbols for the purposes of maintaining DC-balance in a communications system.

Each 8-bit output symbol contains 4 zero bits and 4 one bits, so the code can, like a parity bit, detect all single-bit errors.

The number of binomial coefficient 8-bit patterns with 4 bits set is ${\tbinom {8}{4}}$ = 70. Further excluding the patterns 11110000 and 00001111, this allows 68 coded patterns: 64 data codes, plus 4 additional control codes.

Coding rules

The 64 possible 6-bit input codes can be classified according to their disparity, the number of 1 bits minus the number of 0 bits:

Ones	Zeros	Disparity	Number
0	6	−6	1
1	5	−4	6
2	4	−2	15
3	3	0	20
4	2	+2	15
5	1	+4	6
6	0	+6	1

The 6-bit input codes are mapped to 8-bit output symbols as follows:

• The 20 6-bit codes with disparity 0 are prefixed with 10
  Example: 000111 → 10000111
  Example: 101010 → 10101010
• The 14 6-bit codes with disparity +2, other than 001111, are prefixed with 00
  Example: 010111 → 00010111
• The 14 6-bit codes with disparity −2, other than 110000, are prefixed with 11
  Example: 101000 → 11101000
• The remaining 20 codes: 12 with disparity ±4, 2 with disparity ±6, 001111, 110000, and the 4 control codes, are assigned to codes beginning with 01 as follows:

Type	Input	Output		Type	Input	Output		Complement
−6	000000	01011001		+6	111111	01100110		01_xx__x
−4	000001	01110001		+4	111110	01001110		01xx____
	000010	01110010			111101	01001101
	000100	01100101			111011	01011010		01x____x
	001000	01101001			110111	01010110
	010000	01010011			101111	01101100		01_____xx
	100000	01100011			011111	01011100
−2	110000	01110100		+2	001111	01001011		01____x__
Control	K 000111	01000111		Control	K 111000	01111000
	K 010101	01010101			K 101010	01101010

Because we excluded the patterns 11110000 and 00001111,

Note that no data symbol contains more than four consecutive matching bits, or begins or ends with more than three identical. Thus, the longest run of identical bits that will be produced is 6. (I.e. this is a (0,5) RLL code, with a worst-case running disparity of +3 to -3.)

Any occurrence of 6 consecutive identical bits constitutes a comma sequence or sync mark or syncword; it identifies the symbol boundaries precisely. Those 6 bits straddle the inter-symbol boundary with exactly 3 of those identical bits at the end of one symbol, and 3 of those identical bits at the start of the following next symbol.

See also

• 8b/10b encoding, another fixed-table system with a higher code rate but less error detection.
• 64b/66b encoding, linear feedback register.

External links

• Freepatents online, United States patent 6,876,315: DC-balanced 6B/8B transmission code with local parity.

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