Binary-coded decimal: Difference between revisions

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==Packed BCD==

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InSome computers whose words are multiples of an [[Octet (computing)|octet]] (8-bit byte), for example contemporary IBM mainframe systems, support '''packed BCD''' (or '''packed decimal'''<ref name="Dewar-Smosna_1990"/>) numeric representations, in which each [[nibble]] represents either a decimal digit or a sign.<ref group="nb" name="Packed_chars"/> Packed BCD has been in use since at least the 1960s and is implemented in all IBM mainframe hardware since then. Most implementations are [[big endian]], i.e. with the more significant digit in the upper half of each byte, and with the leftmost byte (residing at the lowest memory address) containing the most significant digits of the packed decimal value. The lower nibble of the rightmost byte is usually used as the sign flag, although some unsigned representations lack a sign flag.

As an example, a 4-byte value consists of 8 nibbles, wherein the upper 7 nibbles store the digits of a 7-digit decimal value, and the lowest nibble indicates the sign of the decimal integer value. Standard sign values are 1100 ([[hexadecimal|hex]] C) for positive (+) and 1101 (D) for negative (−). This convention comes from the zone field for [[EBCDIC]] characters and the [[signed overpunch]] representation.

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=== Disadvantages ===

* Practical existing implementations of BCD are typically slower than operations on binary representations, especially on embedded systems, due to limited processor support for native BCD operations.<ref name="Mathur_1989" />

* Some operations are more complex to implement. [[Adder (electronics)|Adder]]s require extra logic to cause them to wrap and generate a carry early. Also, 15 to 20 per cent% more circuitry is needed for BCD add compared to pure binary.{{Citation needed|date=May 2011}} Multiplication requires the use of algorithms that are somewhat more complex than shift-mask-add (a [[Binary numeral system#Multiplication|binary multiplication]], requiring binary shifts and adds or the equivalent, per-digit or group of digits is required).

* Standard BCD requires four bits per digit, roughly 20 per cent% more space than a binary encoding (the ratio of 4 bits to log₂10 bits is 1.204). When packed so that three digits are encoded in ten bits, the storage overhead is greatly reduced, at the expense of an encoding that is unaligned with the 8-bit byte boundaries common on existing hardware, resulting in slower implementations on these systems.<!-- Could add: encoding or decoding is trivial in software using a table lookup, and fast using direct logic otherwise. In hardware, it requires no more than three gate delays. -->

==Representational variations==

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==={{anchor|TBCD}}Telephony binary-coded decimal (TBCD)===

[[3GPP]] developed '''TBCD''',<ref name="3GPP_2013_TS29002"/> an expansion to BCD where the remaining (unused) bit combinations are used to add specific [[telephony]] characterssymbols,<ref name="ETSI_SPS"/><ref name="OpenSS_XMAP"/> with digits similar to those found in [[DTMF|telephone keypad]]s original design.

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<ref name="Dewar-Smosna_1990">{{cite book |title=Microprocessors - A Programmer's View |author-first1=Robert Berriedale Keith |author-last1=Dewar |author-link1=Robert Berriedale Keith Dewar |author-first2=Matthew |author-last2=Smosna |date=1990 |edition=1 |publisher=[[McGraw-Hill Publishing Company]] |___location=[[Courant Institute]], [[New York University]], New York, USA |isbn=0-07-016638-2 |lccn=89-77320 |page=14}} (xviii+462 pages)</ref>

<ref name="Savard_2018_Decimal">{{cite web |title=Decimal Representations |author-first=John J. G. |author-last=Savard |date=2018 |orig-date=2006 |work=quadibloc |url=http://www.quadibloc.com/comp/cp0203.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180716101321/http://www.quadibloc.com/comp/cp0203.htm |archive-date=2018-07-16}}</ref>

<ref name="Yuen_1977">{{cite journal |title=A New Representation for Decimal Numbers |author-first=Chun-Kwong |author-last=Yuen |journal=[[IEEE Transactions on Computers]] |date=December 1977 |volume=C-26 |issue=12 |doi=10.1109/TC.1977.1674792 |s2cid=40879271 |pages=1286–1288 |url=https://dl.acm.org/doi/10.1109/TC.1977.1674792 |access-date=2020-08-08 |url-status=live |archive-url=https://web.archive.org/web/20200808105531/https://dl.acm.org/doi/10.1109/TC.1977.1674792 |archive-date=2020-08-08|url-access=subscription }}</ref>

<ref name="Kautz_1954">{{cite conference |title=Optimized Data Encoding for Digital Computers |chapter=IV. Examples A. Binary Codes for Decimals, n = 4 |author-last=Kautz |author-first=William H. |author-link=William H. Kautz |conference=Convention Record of the I.R.E., 1954 National Convention, Part 4 - Electronic Computers and Information Theory |publisher=[[I.R.E.]] |series=Session 19: Information Theory III - Speed and Computation |date=June 1954 |___location=Stanford Research Institute, Stanford, California, USA |pages=47–57 [49, 51–52, 57] |url=https://worldradiohistory.com/Archive-IRE/50s/IRE-1954-Part-4-Electronic-Computers-&-Information%20pdf |access-date=2020-07-03 |url-status=live |archive-url=https://web.archive.org/web/20200703180632/https://worldradiohistory.com/Archive-IRE/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information%20pdf |archive-date=2020-07-03 |quote-page=52 |quote=[…] The last column [of Table II], labeled "Best," gives the maximum fraction possible with any code—namely 0.60—half again better than any conventional code. This extremal is reached with the ten heavily-marked vertices of the graph of [[#Kautz|Fig. 4]] for n = 4, or, in fact, with any set of ten code combinations which include all eight with an even (or all eight with an odd) number of "1's." The second and third rows of Table II list the average and peak decimal change per undetected single binary error, and have been derived using the equations of Sec. II for Δ₁ and δ₁. The confusion index for decimals using the criterion of "decimal change," is taken to be c_ij = {{!}}i − j{{!}} &nbsp; i,j = 0, 1, … 9. Again, the "Best" arrangement possible (the same for average and peak), one of which is shown in Fig. 4, is substantially better than the conventional codes. […] Fig. 4 [[#Kautz|Minimum-confusion code for decimals]]. […] δ₁=2 &nbsp; Δ₁=15 […]}} [https://web.archive.org/web/20200703173707/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0049.pdf] [https://web.archive.org/web/20200703175038/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0050.pdf] [https://web.archive.org/web/20200703175214/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0051.pdf] [https://web.archive.org/web/20200703175243/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0052.pdf] [https://web.archive.org/web/20200703175313/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0053.pdf] [https://web.archive.org/web/20200703175344/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0054.pdf] [https://web.archive.org/web/20200703175425/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0055.pdf] [https://web.archive.org/web/20200703175459/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0056.pdf] [https://web.archive.org/web/20200703175529/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0057.pdf] [https://web.archive.org/web/20200703175606/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0058.pdf] [https://web.archive.org/web/20200703175641/https://worldradiohistory.com/hd2/IDX-Site-Technical/Engineering-General/Archive-IRE-IDX/IDX/50s/IRE-1954-Part-4-Electronic-Computers-%26-Information-OCR-Page-0059.pdf] (11 pages) (NB. Besides the combinatorial set of 4-bit BCD "minimum-confusion codes for decimals", of which the author illustrates only one explicitly (here reproduced as [[#Kautz|code I]]) in form of a 4-bit graph, the author also shows a 16-state 4-bit "binary code for analog data" in form of a code table, which, however, is not discussed here. The [[#Kautz II|code II]] shown here is a modification of code I discussed by {{citeref|Berger|1962|Berger|style=plain}}.)</ref>

<ref name="Lippel_1955">{{cite journal |title=A Decimal Code for Analog-to-Digital Conversion |author-last=Lippel |author-first=Bernhard |journal=[[IRE Transactions on Electronic Computers]] |issn=0367-9950 |volume=EC-4 |issue=4 |date=December 1955 |doi=10.1109/TEC.1955.5219487 |pages=158–159}} (2 pages)</ref>

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<ref name="Glixon_1957">{{cite journal |date=March 1957 |title=Can You Take Advantage of the Cyclic Binary-Decimal Code? |author-first=Harry Robert |author-last=Glixon |journal=[[Control Engineering (magazine)|Control Engineering]] |issn=0010-8049 |publisher=[[Technical Publishing Company]], a division of Dun-Donnelley Publishing Corporation, [[Dun & Bradstreet Corp.]] |volume=4 |number=3 |pages=<!-- 3, -->87–91 |url=https://books.google.com/books?id=-_5IAQAAIAAJ}}<!-- https://web.archive.org/web/20180115014809/https://donmooreswartales.com/2010/05/12/harry-glixon/ https://books.google.com/books?id=-_5IAQAAIAAJ&focus=searchwithinvolume&q=cyclic+binary+code --> (5 pages)</ref>

<ref name="White_1953">{{cite journal |title=Coded Decimal Number Systems for Digital Computers |author-first=Garland S. |author-last=White |journal=[[Proceedings of the Institute of Radio Engineers]] |publisher=[[Institute of Radio Engineers]] (IRE) |issn=0096-8390 |eissn=2162-6634 |volume=41 |number=10 |date=October 1953 |doi=10.1109/JRPROC.1953.274330 |s2cid=51674710 |pages=1450–1452}} (3 pages)</ref>

<ref name="Lucal_1959">{{cite journal |author-first=Harold M. |author-last=Lucal |title=Arithmetic Operations for Digital Computers Using a Modified Reflected Binary |journal=[[IRE Transactions on Electronic Computers]] |volume=EC-8 |number=4 |pages=449–458 |date=December 1959 |issn=0367-9950 |doi=10.1109/TEC.1959.5222057 |s2cid=206673385 |url=https://ieeexplore.ieee.org/document/5222057}} (10 pages)</ref>

<ref name="EHub_2015">{{cite web |title=Different Types of Binary Codes |at=Section 2.4 5211 Code |date=2019-05-01 |orig-date=2015-01-28 |work=Electronic Hub |url=https://www.electronicshub.org/disclaimer/ |access-date=2020-08-04 |url-status=live |archive-url=https://web.archive.org/web/20200518203953/https://www.electronicshub.org/disclaimer/ |archive-date=2020-05-18}}</ref>

<ref name="Paul_1995">{{cite web |author-first=Matthias R. |author-last=Paul |title=Unterbrechungsfreier Schleifencode |language=de |trans-title=Continuous loop code |version=1.02 |url=http://www.uni-bonn.de/~uzs180/download/mpbcd102.zip |date=1995-08-10 |orig-date=1994 |access-date=2008-02-11}}{{cbignore}} (NB. The author called this code {{lang|de|Schleifencode}} (English: "loop code"). It differs from [[Gray BCD code]] only in the encoding of state 0 to make it a cyclic [[unit-distance code]] for full-circle rotatory [[slip ring]] applications. Avoiding the all-zero code pattern allows for loop self-testing and to use the data lines for uninterrupted power distribution.)</ref>