excess-3

Biased codes are a way to represent values with a balanced number of positive and negative numbers using a pre-specified number N as a biasing value. Biased codes (and Gray codes) are non-weighted codes. In excess-3 code, numbers are represented as decimal digits, and each digit is represented by four bits as the digit value plus 3 (the "excess" amount):

• The smallest binary number represents the smallest value ({{nobr|0 − excess}}).
• The greatest binary number represents the largest value ({{nobr|2^N+1 − excess − 1}}).

To encode a number such as 127, one simply encodes each of the decimal digits as above, giving (0100, 0101, 1010).

Excess-3 arithmetic uses different algorithms than normal non-biased BCD or binary positional system numbers. After adding two excess-3 digits, the raw sum is excess-6. For instance, after adding 1 (0100 in excess-3) and 2 (0101 in excess-3), the sum looks like 6 (1001 in excess-3) instead of 3 (0110 in excess-3). To correct this problem, after adding two digits, it is necessary to remove the extra bias by subtracting binary 0011 (decimal 3 in unbiased binary) if the resulting digit is less than decimal 10, or subtracting binary 1101 (decimal 13 in unbiased binary) if an overflow (carry) has occurred. (In 4-bit binary, subtracting binary 1101 is equivalent to adding 0011 and vice versa.){{cite book |title=Computer Architecture and Organization |last=Hayes |first=John P. |isbn=0-07-027363-4 |date=1978 |publisher=McGraw-Hill International Book Company |page=156}}

The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented (for subtraction) as easily as a binary number can be ones' complemented: just by inverting all bits. Also, when the sum of two excess-3 digits is greater than 9, the carry bit of a 4-bit adder will be set high. This works because, after adding two digits, an "excess" value of 6 results in the sum. Because a 4-bit integer can only hold values 0 to 15, an excess of 6 means that any sum over 9 will overflow (produce a carry-out).

Another advantage is that the codes 0000 and 1111 are not used for any digit. A fault in a memory or basic transmission line may result in these codes. It is also more difficult to write the zero pattern to magnetic media.

BCD 8-4-2-1 to excess-3 converter example in VHDL:

entity bcd8421xs3 is

port (

a : in std_logic;

b : in std_logic;

c : in std_logic;

d : in std_logic;

an : buffer std_logic;

bn : buffer std_logic;

cn : buffer std_logic;

dn : buffer std_logic;

w : out std_logic;

x : out std_logic;

y : out std_logic;

z : out std_logic

);

end entity bcd8421xs3;

architecture dataflow of bcd8421xs3 is

begin

an <= not a;

bn <= not b;

cn <= not c;

dn <= not d;

w <= (an and b and d ) or (a and bn and cn)

or (an and b and c and dn);

x <= (an and bn and d ) or (an and bn and c and dn)

or (an and b and cn and dn) or (a and bn and cn and d);

y <= (an and cn and dn) or (an and c and d )

or (a and bn and cn and dn);

z <= (an and dn) or (a and bn and cn and dn);

end architecture dataflow; -- of bcd8421xs3

{{Infobox code

|name=3-of-6 extension

|digits=6

|tracks=6

|weight=3

|continuity=No

|cyclic=No

|minimum_distance=2

|maximum_distance=6

|lexicography=1

|complement=(9)

}}

{{Infobox code

|name=4-of-8 extension

|digits=8

|tracks=8

|weight=4

|continuity=No

|cyclic=No

|minimum_distance=4

|maximum_distance=8

|lexicography=1

|complement=9

}}

• 3-of-6 code extension: The excess-3 code is sometimes also used for data transfer, then often expanded to a 6-bit code per CCITT GT 43 No. 1, where 3 out of 6 bits are set.
• 4-of-8 code extension: As an alternative to the IBM transceiver code (which is a 4-of-8 code with a Hamming distance of 2), it is also possible to define a 4-of-8 excess-3 code extension achieving a Hamming distance of 4, if only denary digits are to be transferred.

• Offset binary, excess-N, biased representation
• Excess-128
• Excess-Gray code
• Shifted Gray code
• Gray code
• m-of-n code
• Aiken code

{{Reflist|refs=

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{{cite book |title=Decimal Computation |author-first=Hermann |author-last=Schmid |author-link=Hermann Schmid (computer scientist) |date=1974 |edition=1 |publisher=John Wiley & Sons, Inc. |location=Binghamton, New York, USA |isbn=0-471-76180-X |page=[https://archive.org/details/decimalcomputati0000schm/page/11 11] |url=https://archive.org/details/decimalcomputati0000schm |url-access=registration |access-date=2016-01-03}}

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{{cite book |author=IBM |title=65 Data Transceiver / 66 Printing Data Receiver |date=July 1957 |author-link=IBM}}

{{cite web |title=Binäre Codices |at=Exzeß-3-Code mit Additions- und Subtraktionsverfahren |work=Informations- und Kommunikationstechnik |language=de |author-first=Detlef |author-last=Mietke |location=Berlin, Germany |date=2017 |orig-year=2015 |url=http://elektroniktutor.de/digitaltechnik/codices.html |access-date=2017-04-25 |url-status=live |archive-url=https://web.archive.org/web/20170425181921/http://elektroniktutor.de/digitaltechnik/codices.html |archive-date=2017-04-25}}

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{{cite book |title=Mathematics and Computers |author-first1=George Robert |author-last1=Stibitz |author-link1=George Robert Stibitz |author-first2=Jules A. |author-last2=Larrivee |date=1957 |edition=1 |publisher=McGraw-Hill Book Company, Inc. |publication-place=New York, USA / Toronto, Canada / London, UK |location=Underhill, Vermont, USA |lccn=56-10331 |page=105}} (10+228 pages)

}}

Category:Binary arithmetic

Category:Numeral systems