# Numbers Far Afield |

*by*Phil Tadros

“To study all that’s learnable;

to ship all collected information

to the Creator on the third planet.

That’s the programming.”(Vejur, née Voyager 6, in Star Trek: The Movement Image)

Think about an enormous rising sphere centered on the Earth with a radius of fifteen billion miles, the gap a beam of sunshine travels in a day. That is the *anthroposphere*: the patch of the universe into which humanity and its artifacts have unfold. At its periphery – certainly, defining its periphery – is a one-ton machine hurtling away from us at forty-thousand miles per hour whereas sending radio indicators towards Earth with a broadcast energy of twenty watts. That’s solely a tiny fraction of the wattage utilized by business radio stations on Earth, but in some way Voyager 1, with its puny transmitter, continues to be executing its mission of gathering information about our photo voltaic system and relaying that information again to far-off Earth. And a part of the expertise that makes this feat doable is an algebraic assemble that no one even dreamed of two centuries in the past: the arithmetic of finite fields.

Truly, I shouldn’t be so assured that no one again then dreamed of finite fields; Carl Friedrich Gauss (1777–1855) could have. He was recognized for not writing up his concepts till they’d matured sufficiently, which in some circumstances meant by no means writing them up in any respect.

In his *Disquisitiones Arithmeticae*, revealed in 1801, Gauss launched the thought of two integers being congruent to at least one one other relative to some third integer that he known as the modulus. (When you want a refresher on modular arithmetic, or if it’s new to you, take a look at my essay The Triumphs of Sisyphus.) Simply over thirty years later, in an 1832 monograph, he launched quantity techniques that include not simply the integers but in addition some further numbers thrown in, such because the Gaussian integers, that’s, all numbers of the shape *a* + *bi* the place *i* is the sq. root of −1. Gauss might need puzzled what type modular arithmetic would absorb different quantity techniques like these. If he did pursue this, he by no means reached conclusions that he thought have been value publishing.^{1} However let’s see what sort of modular arithmetic we get once we impose the modulus 2 on the Gaussian integers and on one other comparable quantity system, doing the identical form of lumping-together we did once we seemed on the integers modulo *m*.

**IN SEARCH OF RECIPROCATION**

To gear up, let’s remind ourselves how modular arithmetic works once we impose the modulus 4 on the strange integers. In mod 4 arithmetic, we deal with two numbers as being the identical – or, in technical language, *congruent* – in the event that they differ by a a number of of 4, so for instance 2 occasions 2 is 0. Right here’s (a part of) a quantity line for the integers by which each integer is changed by the rest it leaves once you divide it by 4.

Listed here are the operation tables for addition and multiplication mod 4:

As an example, to make use of the primary desk to compute 2+3 in mod 4 arithmetic, discover the place the place the row marked “2” and the column marked “3” cross; because the entry there’s a 1, we have now 2 + 3 equal to 1 in mod 4 arithmetic, although we frequently use the image ≡ reasonably than the equals signal, with the modulus 4 lurking invisibly behind the “≡”.

On this quantity system, subtraction works superb^{2} however division is an issue. If the difficulty with division have been simply the standard caveat “You possibly can’t divide by 0” it wouldn’t be so dangerous, however on this quantity system you can also’t divide by 2: the un-multiplication drawback “Discover *x* in order that *x* × 2 is 1 mod 4” has no answer within the integers mod 4. That’s, 2 has no reciprocal on this quantity system.

One other four-element quantity system with an analogous defect is the Gaussian integers mod 2, consisting of 0, 1, *i*, and 1+*i* (which I’ll name *j* for brief). Right here’s an image of the Gaussian integers by which each Gaussian integer has been changed by the rest it leaves once you divide it by 2.

I give extra particulars within the Endnotes however right here I’ll simply say that 1+*i* has no reciprocal on this quantity system.^{3}

We get one thing nicer if as an alternative of the Gaussian integers we use the Eisenstein integers. The Eisenstein integers are the advanced numbers of the shape *a*+*bγ* the place *a* and *b* are strange integers and *γ* (that’s what gamma appears like in WordPress, apparently) is the advanced quantity −1/2 + *i* sqrt(3)/2. Don’t let these fractions and that sq. root scare you; all you really want to find out about *γ* is that’s satisfies the equation *γ*^{2}+*γ*+1 = 0, or equivalently, *γ*^{2} = −*γ*−1. After we mod out by 2, we’re solely being attentive to the parity (that’s, oddness or evenness) of *a* and *b*, so we’d as nicely prohibit *a* and *b* to be 0 or 1, in order that the weather of our decreased arithmetic are 0, 1, *γ*, and 1+*γ*; I’ll abbreviate the final of those as *δ*. Right here’s an image of the Eisenstein integers by which each Eisenstein integer has been changed by the rest it leaves once you divide it by 2.

And listed here are the operation tables for addition and multiplication of Eisenstein integers mod 2:

I gained’t clarify all of the entries, however I’ll clarify two of them. To see why *γ* occasions *γ* is *δ*, recall that within the Eisenstein integers not modded out by 2,* γ* occasions *γ* is −1−*γ*. However since −1 and +1 are the identical mod 2, we will rewrite this as 1+*γ*, i.e. *δ*. Likewise, to see why *δ* occasions *δ* is *γ*, apply the distributive regulation to increase 1+*γ* occasions 1+*γ* as 1+2*γ*+*γ*^{2}. Since *γ*^{2} = −1−*γ*, 1+2*γ*+*γ*^{2} turns into 1+2*γ*−1−*γ*, which is simply *γ*.

On this quantity system, each nonzero component has a reciprocal that additionally belongs to the quantity system; particularly, 1 is its personal reciprocal and *γ* and *δ* are reciprocals of one another. Consequently, division works simply superb, the place we carry out division utilizing the rule “To divide by a component, simply multiply by its reciprocal.”

The long-deferred second has arrived once I have to deliver the phrase “area” into the dialog. *Fields* are quantity techniques by which all 4 of the elemental operations of arithmetic – addition, subtraction, multiplication, and division – work properly, no less than so long as you don’t divide by the additive identification component 0.

At any time when *p* is prime, the integers mod *p* type a tidy finite area known as 𝔽_{p}. However these aren’t the one finite fields; as an illustration, we simply met a area with 4 components, specifically the Eisenstein integers mod 2. Equally, for those who take the Gaussian integers mod 3, you may assemble a finite area with 9 components.^{4} The total story is that each time the quantity *n* is an influence of a major, say *n* = *p ^{ok}* with

*p*prime and

*ok*≥ 1, then there’s a finite area

^{5}with

*n*components, written as 𝔽

_{pok}. One technique to assemble finite fields follows the strategy we took with 𝔽

_{4}: begin with an appropriate system of algebraic numbers and mod out by an appropriate prime

*p*. However this isn’t how finite fields have been really invented.

**A RADICAL MATHEMATICIAN**

Finite fields are the creation of the revolutionary mathematician Évariste Galois (1811–1832), who died earlier than his twenty-first birthday. He wasn’t only a mathematical revolutionary; he was additionally a political revolutionary who ardently supported Republican beliefs within the years main as much as the French Revolution of 1830. Political embroilments, unreciprocated affection, and/or suicidal melancholy led him to enter right into a deadly duel (see Why was Évariste Galois killed?). I’ve complained concerning the algebra duels of mid-millennial Italy and the antagonistic impact they’d on the progress of arithmetic, however no less than in these duels, the loser misplaced solely his job, not his life!

Galois was within the concept of equations and extra particularly in a query that had preoccupied algebraists for a lot of a long time: Which algebraic equations will be solved utilizing simply the 4 commonplace operations of algebra (+, −, ×, ÷) together with radicals (sq. roots, dice roots, fourth roots, and many others.)? His novel strategy concerned a form of covert choreography by which the options of an equation swap locations with one another whereas we’re not wanting, and we observe how varied permutations of the options go away some expressions invariant however change the values of different expressions, the best way swapping the 2 roots *r*, *s* of a quadratic equation leaves the values of *r*+*s* and *rs* alone however causes the worth of *r*−*s* to vary signal. It was good work that almost all mathematicians of his period have been unprepared to assimilate, and it’s doable that Galois’ frustration with being misunderstood and unappreciated performed a component within the prepare of occasions that led to his demise.

Though a few of Galois’ work was deemed incomprehensible by his colleagues and denied publication throughout his lifetime, a few of it was revealed. In 1830 he revealed three articles, one entitled “On the speculation of numbers”; it was this text that launched finite fields to the world. Galois’ approach of establishing them is to begin with the integers mod *p* and adjoin a brand new component, in analogy with the best way we constructed the advanced numbers from the actual numbers by adjoining a brand new component *i* satisfying *i*^{2} = −1. Beneath this strategy, we don’t want advanced numbers besides as a supply of inspiration. As an example, to construct the sphere with 4 components, we begin with the two-element area 𝔽_{2} whose solely components are 0 and 1 and we introduce a brand new component known as *α* (that’s an alpha in WordPress, apparently) with the property that *α*^{2} = *α*+1. The place does *α* come from? It doesn’t matter! We assume the existence of such a component and see what penalties the idea results in. If it results in contradictions, nicely, then *α* doesn’t exist. But when the speculation is constant, then *α* exists simply as a lot because it must; we will do math with *α* with out worrying about its ontology. And, simply as letting *i* mix with itself and strange actual numbers by way of addition and multiplication provides rise to the sphere of advanced numbers, letting *α* mix with itself and the weather of 𝔽_{2} by way of addition and multiplication provides rise to the sphere 𝔽_{4}.

Finite fields are sometimes known as Galois fields in honor of their creator, and 𝔽_{pok} is usually written as GF(*p ^{ok}*). One doesn’t normally name the weather of finite fields “numbers”, however they’re definitely near-kindred. To intensify the resemblance, we might rebrand the weather as numbers. As an example, as an alternative of calling the weather of GF(4) 0, 1,

*γ*, and

*δ*, we might name them 0, 1, 2, and three. Then the addition desk would appear like this:

This looks as if a loopy factor to do, since we’re subverting the which means of the symbols 2 and three and altering the best way they add. But there’s hidden sense to this nonsense. In 1901, the mathematician Charles Bouton, learning the sport Nim (which appears to have been impressed by comparable Chinese language take-away video games), realized that the foregoing unusual approach of including pure numbers displays the strategic habits of Nim; in a sure tactical sense, a heap of measurement *a* alongside a heap of measurement *b* acts like a single heap of measurement *c*, supplied that *c* is the same as *a* + *b* for this bizarre type of addition. I wrote extra about Nim in my essay When 1+1 Equals 0. Some individuals, adopting a coinage that I imagine is because of John Conway, discuss with Nim values as “nimbers”.

To see the sample governing that final addition desk, it would assist to interchange 0, 1, 2, and three by their respective base-two representations 00, 01, 10, and 11:

Then the sum of the bit-pair *a*_{1}*a*_{2} and the bit-pair *b*_{1}*b*_{2} is the bit-pair *c*_{1}*c*_{2} the place *c*_{1} is the mod-2 sum of *a*_{1} and *b*_{1} and *c*_{2} is the mod-2 sum of *a*_{2} and *b*_{2}.

If all these 0’s and 1’s are making you consider computer systems, that’s not an accident, as a result of the following scene within the story of finite fields happens within the daybreak of the pc age.

**THE LOST WEEKENDS OF RICHARD HAMMING**

In the summertime of 1946, thirty-one-year-old mathematician Richard Hamming (1915–1998) left the atomic bomb venture at Los Alamos Nationwide Laboratories to take a job at Bell Laboratories, the place a state-of-the-art binary laptop known as the Mannequin V was the satisfaction of the lab. The Mannequin V contained practically ten thousand relays, occupied a thousand sq. toes of flooring house, weighed ten tons, and wasn’t meant for newbies like Hamming. Many questions that him couldn’t be solved by pure thought; he wanted a pc just like the Mannequin V. However Hamming’s low standing meant that essentially the most he might hope for was weekend time on the Mannequin V, and even then, solely when nobody else needed to make use of the machine.

Enter to the Mannequin V used punched paper tape, which isn’t an particularly dependable approach of getting into applications or information into a pc; it’s frequent for a mechanical tape-reader to misinterpret a 1 (a gap within the tape) as a 0 (part of the tape the place a gap might have been however wasn’t) or vice versa. The Mannequin V might additionally malfunction if a speck of mud landed on one among its inside relays. Thankfully the Mannequin V’s designers had included some error-detection functionality into the system. Particularly, all information have been expressed within the type of blocks of 5 bits, of which precisely 3 have been 1’s: the “3-out-of-5 code”. Listed here are the ten allowed blocks, equivalent to the ten decimal digits:

00111 01011 01101 01110 10011 10101 10110 11001 11010 11100

If the pc encountered a block that didn’t have precisely three 1’s, an alarm would sound and the machine’s operator might restart the job, hoping that this time there can be no errors …

… besides that on weekends, when Hamming’s applications have been being run, there was no operator. If the Mannequin V detected an error throughout the weekend, it simply terminated its present job and went on to the following one.

Hamming later recounted:

*Two weekends in a row I got here in and located that each one my stuff had been dumped and nothing was finished. I used to be actually aroused and aggravated as a result of I needed these solutions and two weekends had been misplaced. And so I stated, “Rattling it, if the machine can detect an error, why can’t it find the place of the error and proper it?”*

In 1948, Hamming got here up with the error-correcting code that bears his title. As an alternative of utilizing ten allowed blocks of size 5 because the 3-out-of-5 code does, Hamming’s code used sixteen allowed blocks, or codewords, of size 7:

0000000 0001111 0010110 0011001 0100101 0101010 0110011 0111100

1000011 1001100 1010101 1011010 1100110 1101001 1110000 1111111

This code had the fascinating function that if the tape reader (or a speck of mud in a relay) flipped a single 0 to a 1 in a block, or vice versa, the Mannequin V would have the option not merely to detect the presence of a corrupted bit but in addition to find the corruption and right it by re-flipping the flipped bit again to its authentic worth.

Beneath you’ll see my try at an image of the Hamming code, proven in pink. I’ve used a modified, extra symmetrical^{6} model of Hamming’s code by which the codewords are

1101000 0110100 0011010 0001101 1000110 0100011 1010001 0000000

1011000 0101100 0010110 0001011 1000101 1100010 0110001 1111111

I handled these 16 codewords as factors in a 7-dimensional house after which projected that house into the airplane. You see simply 15 pink dots, however in 7 dimensions there are 16 of them; the 2 codewords 0000000 and 1111111 correspond to the identical level within the airplane on the center of the image. Within the 7-dimensional image there are 128 dots in whole, 16 pink and the remainder black, equivalent to the 128 doable bit-strings of size 7. Two dots of both shade are joined by an edge if the corresponding bit-strings differ in only a single bit.

What makes this code work – that’s, what makes it doable for the pc to right all single-bit errors – is that every black dot has precisely one pink neighbor; that’s, every bit-string that isn’t a codeword will be changed into a codeword by flipping one bit, and this could at all times be finished in exactly a method. It’s exhausting to see what’s happening in such a crowded image, however that’s the worth you pay once you squash a hyper-hyper-hyper-hypercube down into two dimensions!^{7}

To make use of Hamming’s authentic model of his code for information transmission, you chop an extended binary message right into a succession of blocks of size 4. Every of the 16 doable blocks of size 4 is related^{8} with a definite codeword of size 7, and these blocks are transmitted in succession. This introduces an overhead issue of seven/4 and ends in slower transmission, however what we achieve is resilience within the face of channel-noise: so long as at most one bit in every transmitted codeword will get corrupted en path to its vacation spot, the receiver can reconstruct the unique message completely.

Hamming devised different codes of an analogous sort. As an example, there’s the “prolonged Hamming code”, obtained from the 7-bit code by appending to every codeword a bit that’s equal to the mod 2 sum of the opposite 7 bits.^{9} This 8-bit code isn’t simply single-error-correcting; it’s additionally double-error-detecting. That’s, if two bits get corrupted reasonably than only one, the construction of the code allows the receiver to detect the truth that two or extra errors have occurred.

Over the last decade that adopted Hamming’s pioneering work, many individuals rushed in to enhance on Hamming’s codes, particularly within the course of permitting extra robustness within the presence of error. Why not create a code that would right two errors in every block, or three, or extra? And the way low might one cut back the overhead issue?

One essential factor that researchers observed was that the Hamming code admitted a distinct description that used finite fields. Think about the Galois area GF(8) obtained from the 2-element area by adjoining a fictional component *β* satisfying the equation *β*^{3} = *β* + 1. Then the codewords in my symmetrical model of Hamming’s code will be described as exactly these bit-strings *a*_{1}*a*_{2}*a*_{3}*a*_{4}*a*_{5}*a*_{6}*a*_{7} with the property that *a*_{1}*β*^{1} + *a*_{2}*β*^{2} + … + *a*_{7}*β*^{7} = 0 in GF(8).^{10}

Using finite fields helped information researchers towards good codes, which means codes that would right a lot of errors with out incurring an excessive amount of overhead. Probably the greatest early codes primarily based on finite fields was the BCH code of Bose, Ray-Choudhuri, and Hocquenghem.^{11} You realize quick-response codes (aka QR codes), these sq. arrays of black-and-white cells that look meaningless to people however inform a cellphone to navigate to a selected web site? The black and white cells signify 0s and 1s that encode a URL utilizing BCH codes. BCH codes are additionally utilized in CDs, DVDs, and solid-state reminiscence units similar to flash drives.

Hamming, impressed by Eugene Wigner’s well-known 1960 essay “The Unreasonable Effectiveness of Arithmetic within the Pure Sciences”, wrote an essay of his personal in 1980 entitled “The Unreasonable Effectiveness of Arithmetic”. In shortening Wigner’s title, Hamming was broadening the scope of the inquiry to incorporate as an illustration engineering, which strictly talking just isn’t a science. Hamming doesn’t point out error-correcting codes, however he might nicely have mentioned them for instance of the best way concepts developed for unworldly functions can lend themselves to functions unimagined by their creators. Galois invented finite fields in service of the speculation of equations, exploring the symmetries of algebraic expressions for their very own sake. But in some way this purely theoretical analysis grew to become foundational for the transmission and storage of information within the late twentieth century. Talking for the mathematical neighborhood, Hamming wrote: “We have now tried to make arithmetic a constant, stunning factor, and by so doing we have now had a tremendous variety of profitable functions to the actual world.”

However the reverse can also be true: in making an attempt to make issues work in the actual world, utilized mathematicians could discover one thing of timeless magnificence. The prolonged Hamming code turned out to be related to the pure math drawback of how one can most effectively pack spheres in 8-dimensional house – an issue whose final answer by Ukrainian mathematician Maryna Viazovska in 2016 was a significant contributor towards her profitable a Fields Medal in 2022. (The Fields Medal is known as after Canadian mathematician John Charles Fields, no relation to Galois Fields.)

A more sophisticated code than Hamming’s (arising from GF(2^{23})) found by Swiss mathematician Marcel Golay led British mathematician John Leech to find a really dense technique to pack spheres in 24-dimensional house, utilizing spheres centered on a set of factors now known as the Leech lattice, which Viazovska together with others lately confirmed is actually the densest technique to pack spheres in 24 dimensions. Within the late twentieth century, the Leech lattice led to the invention of a number of new sorts of symmetry, extra exactly, the invention of several new finite simple groups. Galois, who gave group concept its title, would have been happy.

The connection between progress in arithmetic and mastery of the pure world isn’t one-way; you would possibly even say it’s reciprocal.

**RITE OF PASSAGE**

If planetary civilizations have a coming-of-age ritual by which they announce their intention of turning into residents of the cosmos, our planet’s bar mitzvah or quinceañera could be stated to have taken place in the summertime of 1977. That was the 12 months we launched Voyager 1 and Voyager 2, not merely to discover our photo voltaic system and dip our toes within the shores of interstellar house but in addition to say howdy to anybody who could be on the market ready for us. Voyager 2 was really launched first, however Voyager 1 took a sooner path to the outer planets and shortly outstripped its twin, increasing the anthroposphere because it went.

When NASA designed the Voyager probes, it confronted many technical challenges, not least of which was the issue of communication. The issue was asymmetrical: we had a lot of alternatives to spice up the power of the sign going from Earth to Voyager, however boosting the power of the sign going within the different course would have required sending a much bigger Voyager and drastically rising the expense of the mission. We had to make use of mathematical smarts rather than battery brawn.

Mission planners had recognized that when Voyager was near Earth, no subtle error-correcting codes can be required. However as soon as Voyager acquired out past Saturn, it was programmed to transmit its messages to Earth utilizing a newly developed kind of BCH code known as a Reed-Solomon code. This code slowed information transmission by an element of 5 – that’s, every little bit of significant information changed into 5 transmitted bits in accordance with the encoding algorithm – however resulted in a dramatic improve in constancy: the bit-error price dropped from over one in a thousand to about one in one million.

Voyager 1 entered interstellar house in the summertime of 2012, lengthy after it had ceased taking beautiful footage of the outer planets and their moons, however nonetheless capable of ship information residence. At such an awesome distance from the Solar, photo voltaic vitality isn’t obtainable; as an alternative, Voyager derives its energy from lumps of plutonium whose decay retains all the things operating. Sooner or later within the coming decade, the radioactive decay will fall under the edge required by Voyager’s circuits; Voyager’s digital mind will stop to operate, and Voyager will turn out to be successfully lifeless. Or maybe it would merely sleep for a number of million years, to be woke up by curious extraterrestrials who discover the craft, decode the Golden Record and different on-board tutorials, and determine to re-power Voyager to see what it would do.

An early scene within the traditional 1968 movie *2001: A Area Odyssey* reveals a monkey bone thrown within the air by an early human; because the bone flies by way of the air, the digital camera jumps ahead in time and reveals us a similarly-shaped spaceship, occupying the identical house on the display screen. With this match-cut Kubrick cinematographically linked humankind’s earliest instruments to applied sciences he might solely dream about in 1968. I’d such as you to think about an analogous match-cut of a extra mathematical nature. We encountered a bone again firstly of the story of Quantity: the Lebombo bone whose creator, for functions we could by no means know, used unary notation to signify the quantity 29. Notches on bones led to counters on tables, marks on paper, and patterns of electrical energy flowing by way of wires, ultimately representing numbers in binary reasonably than unary. So I ask you to image the Lebombo bone morphing into Voyager 1, a lonely emissary of the human race with the arithmetic of finite fields etched into its circuitry, lighting out for unknown territories.

*Because of John Baez, Noam Elkies, Sandi Gubin, Henri Picciotto, and Glen Whitney.*

*In reminiscence of Ellen Propp, 1935–2023*

**ENDNOTES**

#1. The revealed model of *Disquisitiones Arithmeticae* incorporates mysterious references to a nonexistent “Part 8”. A draft of the lacking part was found amongst Gauss’ papers when he died in 1855. Students suppose Gauss could have written the draft as early as 1797. Part 8 treats what Gauss known as the speculation of upper congruences. He checked out addition and multiplication of polynomials by which all of the coefficients get decreased mod* p*, for some mounted prime *p*; as an illustration, 1+*x* occasions 1+*x* mod 2 turns into 1+*x*^{2} because the intervening time period 2*x* will get decreased to 0*x* or 0. Gauss was concerned with polynomials which might be irreducible in mod *p* arithmetic; that’s, polynomials that may’t be factored the best way 1 + *x*^{2} will be factored as 1+*x* occasions 1+*x* in mod 2 arithmetic. Gauss found out a technique to rely the degree-*ok* polynomials which might be irreducible in mod-*p* arithmetic, and a consequence of his work is that there’s at all times no less than one such polynomial, for every selection of *p* and *ok*. In a while, it was realized by others that every such polynomial provides a technique to assemble a finite area with *p ^{ok}* components, so you might say that Gauss proved the existence of all of the finite fields with out realizing what finite fields have been!

#2. As an example, to compute 1 minus 3, or equivalently, to unravel *x* + 3 ≡ 1, you take a look at all of the values within the desk under the column-heading “3”, that are all of the values of *x* + 3 as *x* varies from 0 to three, to see if one among them matches the specified worth 1.

We discover a 1 within the second-to-last row. Success! Because the row-heading is 2, we study that 1 minus 3 is 2. This simply restates the truth that 2 plus 3 is 1, which we already knew.

#3. The Gaussian integers are the advanced numbers of the shape *a* + *bi* the place *a* and *b* are strange integers and *i* is the sq. root of −1. After we mod out by 2, we’d as nicely solely enable *a* and *b* to be 0 or 1. So as to add or multiply two components of this miniature quantity system, add or multiply them within the strange approach, after which simply take the remainders that outcome when the actual and imaginary elements get divided by 2. As an example, 1 + *i* plus* i* would ordinarily be 1 + 2*i*, however once you take remainders, you get 1 + 0*i*, or 1. Likewise, *i* occasions *i* would ordinarily be −1, however mod 2 that turns into 1. Listed here are the operation tables for addition and multiplication of Gaussian integers mod 2, the place I’ve abbreviated 0 + 0*i*, 1+0*i*, 0+1*i*, and 1+1*i* as 0, 1, *i*, and *j*:

On this quantity system, subtraction works superb however division is as soon as once more an issue: the final row of the multiplication desk (the one which multiplies *j* by all the things in sight) doesn’t include 1 as an entry, which signifies that the component *j* (alias 1+*i*) has no reciprocal.

#4. Listed here are a few footage associated to the finite area with 9 components.

The primary image reveals the start of the infinite spiral generated by the powers of 1+*i* within the advanced airplane. To avoid wasting house, I once more write 1+*i* as *j*. The advanced quantity 0 is on the heart of the sq. marked 0. To its proper is a sq. marked *j*^{0} whose heart is the advanced quantity *j*^{0} = 1. Above that may be a sq. marked *j*^{1} whose heart is the advanced quantity *j*^{1} = 1+*i*. The spiral continues in a counterclockwise course, with increased powers of *j* progressing outward with successive angles of 45 levels. I’ve divided the airplane into 3-by-3 squares, with 0 on the heart of its 3-by-3 sq.. Two Gaussian integers are congruent mod 3 supplied they lie in the identical relative positions throughout the 3-by-3 squares they occupy.

The second image reveals what occurs once you stack all these 3-by-3 squares on high of the one which incorporates 0. This implies mapping the advanced quantity *a* + *bi* to the advanced quantity *a*′ + *b*′*i*, the place *a*′ and *b*′ are the integers between −1 and 1 which might be congruent mod 3 to *a* and *b* respectively. Now the unbounded rising spiral turns into a repeating path that jumps between the eight small 1-by-1 squares adjoining to the central 1-by-1 sq.. Multiplying one thing by 1 + *i* mod 3 simply means following the arrows.

#5. In actual fact, any two finite fields with the identical variety of components are isomorphic to at least one one other; every is only a relabeling of the opposite. Mathematicians say “There is only one area of order *p ^{ok}* as much as isomorphism.” The phrase “as much as” is utilized in a lot the identical approach because the phrase “modulo”.

#6. This code has cyclic symmetry: for those who drop the final little bit of a codeword and stick it on the entrance, what you get can also be a codeword. Such codes are known as *cyclic codes*, they usually performed a significant function within the early days of coding concept.

#7. The *Hamming distance* between two bit-strings of size seven is the variety of bits it is advisable flip so as to convert the primary bit-string into the second bit-string. Equivalently, it’s the variety of positions *n* between 1 and seven such that the *n*th little bit of the primary bit-string differs from the *n*th little bit of the second bit-string. If we mannequin these bit-strings as vertices of the 7-cube, then the Hamming distance is the gap a bug would wish to journey to get from one vertex to the opposite if the bug can solely journey alongside edges of the 7-cube. The Hamming code has the property that each bit-string is both a codeword or has Hamming distance 1 from a codeword.

#8. Given a block *a*_{1}*a*_{2}*a*_{3}*a*_{4} of size 4, the transmitted block *b*_{1}*b*_{2}*b*_{3}*b*_{4}*b*_{5}*b*_{6}*b*_{7} is set by the formulation

the place +_{2} signifies mod 2 addition. Right here* b*_{1}, *b*_{2}, and *b*_{4} are known as test bits; if there isn’t a noise on the communication channel, then they supply no info not already current in *b*_{3}, *b*_{5}, *b*_{6}, and *b*_{7}, that are simply the unique message bits *a*_{1}, *a*_{2}, *a*_{3}, and* a*_{4}, however the redundancy makes it doable for the receiver of the data to detect and proper errors. If we bundle the unique bits *a*_{1} , … , *a*_{4} within the 1-by-4 matrix

and the encoded bits *b*_{1}, … , *b*_{7} within the 1-by-7 matrix

then we will use matrix algebra to jot down *B* because the product of the matrix *A* and the 4-by-7 matrix

Right here I’m utilizing a slight variant of the notion of matrix multiplication alluded to in What is a matrix?, however with one twist: the place the traditional definition of matrix multiplication has us do strange addition and strange multiplication of numbers, right here we carry out mod 2 addition and mod 2 multiplication as an alternative.

#9. Equivalently and extra symmetrically, we could say that the code transmits the 4 authentic bits *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4} together with 4 test bits, every fashioned by taking the mod-2 sum of three of the 4 authentic bits.

#10. As an example, contemplate the codeword 1101000, with 1’s within the 1st, 2nd, and 4th positions. This corresponds to the component *β*^{1} + *β*^{2} + *β*^{4} in GF (8). To see if it equals the 0 component of GF(8), first compute *β*^{4} by multiplying the equation *β*^{3} = *β*+1 by *β*, acquiring *β*^{4} = *β*^{2} + *β*. Then we discover that *β*^{1} + *β*^{2} + *β*^{4} is *β* + *β*^{2} + (*β*^{2} + *β*), and all phrases cancel, giving 0.

#11. It ought to really be known as the BRH code, because the surname of Bose’s collaborator was Ray-Choudhuri, not Choudhuri.

**REFERENCES**

Kenneth Andrews et al., The Development of Turbo and LDPC Codes for Deep-Space Applications

John Baez, Golay Code

Frédéric Brechenmacher, A History of Galois fields

Paul Ceruzzi, Bell Labs Relay Computers, ACM Digital Library, Encyclopedia of Laptop Science

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, The sphere packing problem in dimension 24, Annals of Arithmetic (2) 185 (2017), no. 3, 1017–1033

Gunther Frei, The Unpublished Section Eight: On the Way to Function Fields over a Finite Field

Richard Hamming, The Unreasonable Effectiveness of Mathematics, reprinted from The American Mathematical Month-to-month, Vol. 87, No. 2. (Feb., 1980), pp. 81-90

Roger Ludwig and Jim Taylor, Voyager Telecommunications

Heinz Lüneburg, On the Early History of Galois Fields, Finite Fields and Purposes, pp. 341–355

NASA Jet Propulsion Laboratory, Images Voyager Took

Thomas Thompson, From Error-Correcting Codes Via Sphere Packings to Easy Finite Teams

Maryna Viazovska, The sphere packing problem in dimension 8, Annals of Arithmetic (2) 185 (2017), no. 3, 991–1015

Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, reprinted from Communications in Pure and Utilized Arithmetic, Vol. 13, No. I (February 1960)