Yet Another Mathblog » sage http://wdjoyner.wordpress.com Wed, 19 Aug 2009 01:10:10 +0000 http://wordpress.com/ en hourly 1 http://www.gravatar.com/blavatar/b1a6a6d87604ac2356ff5c313eeb0704?s=96&d=http://s.wordpress.com/i/buttonw-com.png Yet Another Mathblog » sage http://wdjoyner.wordpress.com The enlightenment of Professor Bigglesnot http://wdjoyner.wordpress.com/2009/08/18/the-enlightenment-of-professor-bigglesnot/ http://wdjoyner.wordpress.com/2009/08/18/the-enlightenment-of-professor-bigglesnot/#comments Wed, 19 Aug 2009 00:24:46 +0000 wdjoyner http://wdjoyner.wordpress.com/?p=93 ]]>

An enthusiastic “Yippee!” echoed down the corridor. so loud it woke several faculty members in
nearby offices. Some even got up out of their chairs and looked up and down the hallway before
returning to grading or research or freecell before falling asleep again. But Bigglesnot was excited.
After all, computing automorphism semigroups of quantum hyperalgebras was his life’s passion, ever since he was a graduate student. In front of him, was the latest issue of the Quantum Hyperalgebra Journal, newly released from its plastic shrink-wrap. It was opened to the article which was the focus of Bigglesnot’s attention – the esteemed Ziggotwat’s discussion of a new algorithm to compute automorphism semigroups of quantum hyperalgebras. Bigglesnot could see immediately from the tables of new data presented that Ziggotwat’s implementation was faster, better and more general than his own program. Whispering “Awesome! Awesome! So, awesome! …” under his breath, he shot off an email to Ziggotwat asking for more information, and, if at all possible, further details on the implementation. Could he please post or email the code, for others to look at? That would be awesome!

Days and weeks went by, but with no reply. One morning B found an email from Z: “…the code needs to be cleaned up first …”, “… so sorry for the late reply …”, but “.. thanks for your interest!”, was the gist. As luck would have it, B spotted Z a few weeks later at the annual meeting of the Society of Quantum Hyperalgebraists. Undaunted, one night after the talks of the meeting were finished, B bombarded Z with free beer and flattery peppered with questions about his program. “In fact”, Z finally confessed, “all the work was done by my former student Pipperpop, who has graduated and does not reply to my emails. I can send you what I have – but no promises!” Drunk, but now estatic, Bigglesnot managed to say “Awesome!” before he fell off his barstool.

In the months that followed, B pored through the incomplete, undocumented code. It was provided as a sequence of files, each one seeming dependent on another. They wouldn’t compile, no matter how B tried. Each day for a month, after attending to his classes, B would try to modify one of the files, hoping that a small change would allow him to compile the files into a functioning program. Each day, he would draft an email to Z (or to P, or to the editor of QHJ) asking what kind of LSD was he tripping on to make him high enough to think this mess of code would ever compile?
And each day, he wisely deleted the draft. Bigglesnot, usually filled to overflowing with self-confidence, was defeated.

Finally, B realized the solution! Not the solution of to how to compile Pipperpop’s poop, but the solution to the general problem. Software computations submitted to scientific journals must be “open” – if scientific data obtained as a result of a software computation is part of research paper submitted for publication then the source code for that software must also be made public and verified before publication. Enlightened, Bigglesnot was optimistic once again.

This postwas inspired by the excellent paper: Pederson, Empiricism is Not a Matter of Faith, Computational Linguistics, Volume 34, Number 3, pp. 465-470, September 2008.
http://www.d.umn.edu/~tpederse/full-pubs.html

]]> http://wdjoyner.wordpress.com/2009/08/18/the-enlightenment-of-professor-bigglesnot/feed/ 0 wdjoyner Sage at the NSF-CDI+ECCAD conferences http://wdjoyner.wordpress.com/2009/05/07/sage-at-the-nsf-cdieccad-conferences/ http://wdjoyner.wordpress.com/2009/05/07/sage-at-the-nsf-cdieccad-conferences/#comments Thu, 07 May 2009 14:19:43 +0000 wdjoyner http://wdjoyner.wordpress.com/?p=58 ]]>

This is a very quick view of some highlights of the conference http://www4.ncsu.edu/~kaltofen/NSF_WS_ECCAD09_Itinerary.html. I think further details of the talks will appear on the conference webpage later. This is very incomplete – just a few thought I was able to
jot down correctly.

Panel discussion:
Q1: What are the grand challenges of symbolic computing?
Is the term “symbolic computation” to broad? (Hybrid symbolic/numerical, algebraic geometric computation, algebraic combinatorial/group-theoretical, computer proofs, tensor calculations, differential, mathematical knowledge/database research, user interfaces, …)

General ans: No. Hoon Hong points out that user interfaces are lower level but below to the same group.

Q2: How can the latest algorithmic advances be made readily available: google analog of problem formulation? (Idea: suppose someone has a clever idea for a good algorithm but not enough discipline to implement it …)

One answer: Sage can put software together – is this the right way? Analog of stackoverflow.com?

Q3: What is the next killer app for symbolic computation? (Oil app of Groebner bases, cel phone app, robotics, …)

Q4: Can academically produced software such as LAPACK, LINBOX, SAGE compete with commercial software?

Hoon Hong answer: Yes but why? Why not cooperate. Support Sage very much but more research on interfaces and integration of different systems could lead to cooperation of the commercial systems with Sage.

Another panel:
Q: What are the spectacular successes and failures of computer algebra?

Failures:
(a) Small number of researchers.
(b) Sage could fail from lack of lies with the symbolic/numerical community (as Maxima/Axiom did). Matlab may fail due to uphill battle to integrate Mupad into good symbolic toolbox. (Many voiced view that Matlab is strong because of its many toolboxes, on the panel and privately.)
(c) Education at the High School level using CA.
(d) Presenting output of CA intelligently and in a standard format.
(e) Failure to teach people how to properly use a CA system.

Successes:
(a) Sage – interesting new effort (with caveat above)
(b) Groebner bases, LLL.

My talk on Sage raised a lot of questions. My There is both strong support for Sage and some questions on its design philisophy. My page 6 at http://sage.math.washington.edu/home/wdj/expository/nsf-eccad2009/
was a source of lots of questions.

At ECCAD http://www.cs.uri.edu/eccad2009/Welcome.html, Sage was mentioned a few times in talks as well as in some posters. The “main” Sage event was a laptop software demo Which Karl-Dieter Crisman set up for Sage.

Overall, a good experience for Sage.

]]> http://wdjoyner.wordpress.com/2009/05/07/sage-at-the-nsf-cdieccad-conferences/feed/ 0 wdjoyner Sage at AMS-MAA 2009 Washington DC http://wdjoyner.wordpress.com/2009/01/08/sage-at-ams-maa-2009-washington-dc/ http://wdjoyner.wordpress.com/2009/01/08/sage-at-ams-maa-2009-washington-dc/#comments Fri, 09 Jan 2009 03:40:35 +0000 wdjoyner http://wdjoyner.wordpress.com/2009/01/08/sage-at-ams-maa-2009-washington-dc/ ]]>

I think Sage gained a lot of publicity this year both by being at
the booth but also having an MAA panel discussion and an AMS session.
The panel discussion was good to be able to meet others in the
teaching community. I think this is related to Sage development because
projects like the educational open source software webworks has a
funding model which seems to be successful. I think Karl Crisman said he
would try to follow up on that.

A few people I met at the booth said they were interested in Sage development
but more stopped by saying that either they or their students could not afford
Maple or Mma and was looking into a cheaper quality alternative. The collaroration
possibilities of the Sage server was a strong “selling point” for smaller schools
which could load sage on a webserver.

There were some really good talks at the Sage session. For example,
Marshall’s talk had amazing graphics and Robert Miller’s talk was very well attended
(with maybe twice as many people in the audience as some of the others).
I thought the quality overall was great, but I’m very partial to such topics of course.

The general message I got from many was that more written material
on Sage in use would be welcomed, especially books. I was touched by one guy
who explained to me that his students were very poor (waitresses, for example)
who cannot afford calculus texts and commercial math programs. The point
he was implicitly making was that by offering software and documentation for
free we are actually improving the quality of such peoples’ lives in a real way.

]]> http://wdjoyner.wordpress.com/2009/01/08/sage-at-ams-maa-2009-washington-dc/feed/ 2 wdjoyner Future coding theory in Sage projects http://wdjoyner.wordpress.com/2008/12/11/future-coding-theory-in-sage-projects/ http://wdjoyner.wordpress.com/2008/12/11/future-coding-theory-in-sage-projects/#comments Thu, 11 Dec 2008 05:32:05 +0000 wdjoyner http://wdjoyner.wordpress.com/?p=53 ]]>
Here are a few ideas for future Sage projects in error-correcting codes.

Needed in Sage is

  • a rewrite in Cython, for speed,
  • special decoding algorithms, also in Cython,
  • for special decoders, it seems to me that one either needs
  1. more classes (eg, a CyclicCodes class), each of which will have a decode method, or
  2. another attribute, such as a name (string) which can be used to determine which decoder method to use.
  • codes over Rings (Cesar Agustin Garcia Vazquez is working on this)
  • codes defined from finite groups fings – for example split group codes,
  • a fast minimum distance function (followed by a fast weight distribution function), valid for all characteristics.

It seems more “Pythonic” to add more classes for decoders, but I am not sure.

]]> http://wdjoyner.wordpress.com/2008/12/11/future-coding-theory-in-sage-projects/feed/ 2 wdjoyner Steiner systems and codes http://wdjoyner.wordpress.com/2008/10/02/steiner-systems-and-codes/ http://wdjoyner.wordpress.com/2008/10/02/steiner-systems-and-codes/#comments Thu, 02 Oct 2008 12:39:58 +0000 wdjoyner http://wdjoyner.wordpress.com/?p=47 ]]>

A t-(v,k,λ)-design D=(P,B) is a pair consisting of a set P of points and a collection B of k-element subsets of P, called blocks, such that the number r of blocks that contain any point p in P is independent of p, and the number λ of blocks that contain any given t-element subset T of P is independent of the choice of T. The numbers v (the number of elements of P), b (the number of blocks), k, r, λ, and t are the parameters of the design. The parameters must satisfy several combinatorial identities, for example:

\lambda _i = \lambda \left(\begin{array}{c} v-i\\ t-i\end{array}\right)/\left(\begin{array}{c} k-i\\ t-i\end{array}\right)

where \lambda _i is the number of blocks that contain any i-element set of points.

A Steiner system S(t,k,v) is a t-(v,k,λ) design with λ=1. There are no Steiner systems known with t>5. The ones known (to me anyway) for t=5 are as follows:

S(5,6,12), S(5,6,24), S(5,8,24), S(5,7,28), S(5,6,48), S(5,6,72), and S(5,6,84).

Question: Are there others?

A couple of these are well-known to arise as the support of codewords of a constant weight in a linear code C (as in the Assmus-Mattson theorem, discussed in another post) in the case when C is a Golay code (S(5,6,12) and S(5,8,24)). See also the wikipedia entry for Steiner system.

Question: Do any of these others arise “naturally from coding theory” like these two do? I.e., do they all arise as the support of codewords of a constant weight in a linear code C via Assmus-Mattson?

Here is a Sage example to illustrate the case of S(5,8,24):

sage: C = ExtendedBinaryGolayCode()
sage: C.assmus_mattson_designs(5)
['weights from C: ',
[8, 12, 16, 24],
‘designs from C: ‘,
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]],
‘weights from C*: ‘,
[8, 12, 16],
‘designs from C*: ‘,
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]
sage: C.assmus_mattson_designs(6)
0
sage: blocks = [c.support() for c in C if hamming_weight(c)==8]; len(blocks)
759

]]> http://wdjoyner.wordpress.com/2008/10/02/steiner-systems-and-codes/feed/ 0 wdjoyner The Assmus-Mattson Theorem, Golay codes, and Mathieu groups http://wdjoyner.wordpress.com/2008/05/25/the-assmus-mattson-theorem-golay-codes-and-mathieu-groups/ http://wdjoyner.wordpress.com/2008/05/25/the-assmus-mattson-theorem-golay-codes-and-mathieu-groups/#comments Sun, 25 May 2008 20:50:07 +0000 wdjoyner http://wdjoyner.wordpress.com/?p=22 ]]>

A block design is a pair (X,B), where X is a non-empty finite set of v>0 elements called points, and B is a non-empty finite multiset of size b whose elements are called blocks, such that each block is a non-empty finite multiset of k points. A design without repeated blocks is called a simple block design. If every subset of points of size t is contained in exactly \lambda blocks the the block design is called a t–(v,k,\lambda) design (or simply a t-design when the parameters are not specfied). When \lambda = 1 then the block design is called a S(t,k,v) Steiner system.

Let C be an [n,k,d] code and let C_i = \{ c \in C\ |\ wt(c) = i\} denote the weight i subset of codewords of weight i. For each codeword c\in C, let supp(c)=\{i\ |\ c_i\not= 0\} denote the support of the codeword.

The first example below means that the binary [24,12,8]-code C has the property that the (support of the) codewords of weight 8 (resp, 12, 16) form a 5-design.

Example: Let $C$ denote the extended binary Golay code of length 24. This is a self-dual [24,12,8]-code. The set X_8 = \{supp(c)\ |\ c \in C_8\} is a 5-(24, 8, 1) design; X_{12} = \{supp(c)\ |\ c \in C_{12}\} is a 5-(24, 12, 48) design;and, X_{16} = \{supp(c)\ |\ c \in C_{16}\} is a 5-(24, 16, 78) design.

This is a consequence of the following theorem of Assmus and Mattson.

Assmus and Mattson Theorem (section 8.4, page 303 of [HP]):

Let A_0, A_1, ..., A_n be the weight distribution of the codewords in a binary linear [n , k, d] code C, and let A_0^\perp, A_1^\perp, ..., A_n^\perp be the weight distribution of the codewords in its dual [n,n-k, d^\perp] code C^\perp. Fix a t, 0<t<d, and let s = |\{ i\ |\ A_i^\perp \not= 0, 0<i\leq n-t\, \}|.
Assume s\leq d-t.

  • If A_i\not= 0 and d\leq i\leq n then C_i = \{ c \in C\ |\ wt(c) = i\} holds a simple t-design.
  • If A_i^\perp\not= 0 and d^\perp\leq i\leq n-t then C_i^\perp = \{ c \in C^\perp \ |\ wt(c) = i\} holds a simple t–design.
  • If A_i^\perp\not= 0 and d^\perp\leq i\leq n-t then C_i^\perp = \{ c \in C^\perp \ |\ wt(c) = i\} holds a simple t–design.

In the Assmus and Mattson Theorem, X is the set \{1,2,...,n\} of coordinate locations and B = \{supp(c)\ |\ c \in C_i\} is the set of supports of the codewords of C of weight i. Therefore, the parameters of the t-design for C_i are

  • t = given,
  • v = n,
  • k = i, (this k is not to be confused with dim(C)!),
  • b = A_i,
  • \lambda = b*binomial(k,t)/binomial(v,t)

(by Theorem 8.1.6, p. 294, in \cite{HP}). Here is a SAGE example.


sage: C = ExtendedBinaryGolayCode()
sage: C.assmus_mattson_designs(5)
['weights from C: ',
[8, 12, 16, 24],
'designs from C: ',
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]],
'weights from C*: ',
[8, 12, 16],
'designs from C*: ',
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]
sage: C.assmus_mattson_designs(6)
0

The automorphism group of the extended binary Golay code is the Mathieu group M_{24}. Moreover, the code is spanned by the codewords of weight 8.

References:
[HP] W. C. Huffman, V. Pless, Fundamentals of error-correcting codes, Cambridge Univ. Press, 2003.
[CvL] P. Cameron, J. van Lint, Graphs, codes and designs, Cambridge Univ. Press, 1980.

]]> http://wdjoyner.wordpress.com/2008/05/25/the-assmus-mattson-theorem-golay-codes-and-mathieu-groups/feed/ 0 wdjoyner On a conjecture of Goppa on codes and counting points hyperelliptic curves http://wdjoyner.wordpress.com/2008/04/26/on-a-conjecture-of-goppa-on-codes-and-counting-points-hyperelliptic-curves/ http://wdjoyner.wordpress.com/2008/04/26/on-a-conjecture-of-goppa-on-codes-and-counting-points-hyperelliptic-curves/#comments Sat, 26 Apr 2008 05:27:43 +0000 wdjoyner http://wdjoyner.wordpress.com/?p=18 ]]>

What is the best code of a given length? This natural, but very hard, question motivates the following definition.

Definition: Let B_q(n,d) denote the largest k such that there exists a [n,k,d]-code in GF(q)^n.

Determining B_q(n,d) is one of the main problems in the theory of error-correcting codes.

Let C_i be a sequence of linear codes with parameters [n_i,k_i,d_i] such that n_i\rightarrow \infty as i \to \infty, and such that both the limits R=\lim_{i\to \infty}\frac{k_i}{n_i} and \delta=\lim_{i\rightarrow \infty}\frac{d_i}{n_i} exist. The Singleton (upper) bound implies R+\delta\leq 1.

Let \Sigma_q denote the set of all (\delta,R)\in [0,1]^2 such that there exists a sequence C_i, i=1,2,..., of [n_i,k_i,d_i]-codes for which \lim_{i\to \infty} d_i/n_1=\delta and \lim_{i\rightarrow \infty} k_i/n_i=R.

Theorem (Manin [SS]): There exists a continuous decreasing function \alpha_q:[0,1]\to [0,1], such that

  1. \alpha_q is strictly decreasing on [0,\frac{q-1}{q}],
  2. \alpha_q(0)=1,
  3. if {\frac{q-1}{q}}\leq x\leq 1 then \alpha_q(x)=0,
  4. \Sigma_q=\{(\delta,R)\in [0,1]^2\ |\ 0\leq R\leq \alpha_q(\delta)\}.

Theorem (Gilbert-Varshamov): We have \alpha_q(x)\geq 1- x\log_q(q-1)-x\log_q(x)-(1-x)\log_q(1-x). In other words, for each fixed \epsilon >0, there exists an [n,k,d]-code C (which may depend on \epsilon) with R(C)+\delta(C)\geq 1- \delta(C)\log_q({\frac{q-1}{q}})-\delta(C)\log_q(\delta(C))-(1-\delta(C))\log_q(1-\delta(C))-\epsilon. In particular, in the case q=2, \alpha_q(x)\geq1-H_2(\delta), where H_q(\delta)=\delta\cdot \log_q(q-1)-\delta\log_q(\delta)-(1-\delta)\log_q(1-\delta) is the entropy function (for a q-ary channel).

Plot of the Gilbert-Varshamov (dotted), Elias (red), Plotkin (dashed), Singleton (dash-dotted), Hamming (green), and MRRW (stepped) curves using SAGE:

asymptotic coding theory bounds
Not a single value of \alpha_q(x) is known for 0<x<{\frac{q-1}{q}}!

Goppa’s conjecture: \alpha_q(x) = 1-H_2(\delta). (In other words, the Gilbert-Varshamov lower bound is actually attained in the case q=2.)

Though my guess is that Goppa’s conjecture is true, it is known (thanks to work of Zink, Vladut and Varshamov) that the q-ary analog of Goppa’s conjecture is false for q>49. On the other hand, we have the following theorem [J].

Theorem: If B(1.77,p) is true for infinitely many primes p with p\equiv 1\pmod 4 then Goppa’s conjecture is false.

The rest of this post will be devoted to explaining the notation in this Theorem. The notation B(c,p) means: “For all subsets S\subset GF(p), |X_S(GF(p))| \leq c\cdot p holds.” The notation X_S means: For each non-empty subset S\subset GF(p), define the hyperelliptic curve X_S by y^2=f_S(x), where f_S(x) = \prod_{a\in S}(x-a).

It may seem strange that a statement about the asymptotic bounds of error-correcting curves is related to the number of points on hyperelliptic curves, but for details see [J]. Unfortunately, it appears much more needs to be known about hyperelliptic curves over finite fields if one is to try to use this connection to prove or disprove Goppa’s conjecture.

References:
[J] D. Joyner, On quadratic residue codes and hyperelliptic curves. Discrete Mathematics & Theoretical Computer Science, Vol 10, No 1 (2008).
[SS] S. Shokranian and M.A. Shokrollahi, Coding theory and bilinear complexity, Scientific Series of the International Bureau, KFA Julich Vol. 21, 1994.

Here’s how the above plot is constructed using SAGE:


sage: f1 = lambda x: gv_bound_asymp(x,2)
sage: P1 = plot(f1,0,1/2,linestyle=":")
sage: f2 = lambda x: plotkin_bound_asymp(x,2)
sage: P2 = plot(f2,0,1/2,linestyle="--")
sage: f3 = lambda x: elias_bound_asymp(x,2)
sage: P3 = plot(f3,0,1/2,rgbcolor=(1,0,0))
sage: f4 = lambda x: singleton_bound_asymp(x,2)
sage: P4 = plot(f4,0,1/2,linestyle="-.")
sage: f5 = lambda x: mrrw1_bound_asymp(x,2)
sage: P5 = plot(f5,0,1/2,linestyle="steps")
sage: f6 = lambda x: hamming_bound_asymp(x,2)
sage: P6 = plot(f6,0,1/2,rgbcolor=(0,1,0))
sage: show(P1+P2+P3+P4+P5+P6)

If you are interested in computing what c might be in the above statement B(c,p), the following SAGE code may help.
First, the Python function below will return a random list of elements without repetition, if desired) from an object. 

def random_elements(F,n,allow_repetition=True):
    """
    F is a SAGE object for which random_element is a method
    and having >n elements. For example, a finite field. Here
    n>1 is an integer. The output is to have no repetitions
    if allow_repetion=False.
    EXAMPLES:
        sage: p = nth_prime(100)
        sage: K = GF(p)
        sage: random_elements(K,2,allow_repetition=False)
        [336, 186]
    The command random_elements(K,p+1,allow_repetition=False)
    triggers a ValueError.
    """
    if (n>F.order()-2 and allow_repetition==False):
        raise ValueError("Parameters are not well-choosen.")
    L = []
    while len(L)<n:
        c = F.random_element()
        if allow_repetition==False:
            if not(c in L):
                L = L+[c]
        else:
            L = L+[c]
    return L

Next, the SAGE code computes |X_S(GF(p))|/p for a particular choice of the parameters. This gives a lower bound for c.


sage: p = nth_prime(100)
sage: K = GF(p)
sage: R. = PolynomialRing(K, "x")
sage: S = random_elements(K,ZZ((p+1)/2),allow_repetition=False)
sage: fS = prod([x - a for a in S])
sage: XS = HyperellipticCurve(fS)
sage: RR(len(XS.points())/p)
0.975970425138632

Based on experiments with this code, it appears that for “random” chices of S, and p “large”, the quotient |X_S(GF(p))|/p is very close to 1.

]]> http://wdjoyner.wordpress.com/2008/04/26/on-a-conjecture-of-goppa-on-codes-and-counting-points-hyperelliptic-curves/feed/ 0 wdjoyner asymptotic coding theory bounds “Circle decoding” of the [7,4,3] Hamming code http://wdjoyner.wordpress.com/2008/04/24/circle-decoding-of-the-743-hamming-code/ http://wdjoyner.wordpress.com/2008/04/24/circle-decoding-of-the-743-hamming-code/#comments Thu, 24 Apr 2008 14:51:08 +0000 wdjoyner http://wdjoyner.wordpress.com/?p=16 ]]>

This is a well-known trick but I’m posting it here since I often have a hard time tracking it down when I need it for an introductory coding theory talk or something. Hopefully someone else will find it amusing too!

Let F = GF(2) and let C be the set of all vectors in the third column below (for simplicity, we omit commas and parentheses, so 0000000 is written instead of (0,0,0,0,0,0,0), for example).

The [7,4,3] Hamming code
  decimal     binary     Hamming [7,4] 
0 0000 0000000
1 0001 0001110
2 0010 0010101
3 0011 0011011
4 0100 0100011
5 0101 0101101
6 0110 0110110
7 0111 0111000
8 1000 1000111
9 1001 1001001
10 1010 1010010
11 1011 1011100
12 1100 1100100
13 1101 1101010
14 1110 1110001
15 1111 1111111

This is a linear code of length 7, dimension 4, and minimum distance 3. It is called the Hamming [7,4,3]-code.
In fact, there is a mapping from F^4 to C given by \phi(x_1,x_2,x_3,x_4)={\bf y}, where
{\bf y}= \left( \begin{array}{c} y_1 \\ y_2 \\ y_3 \\ y_4 \\ y_5 \\ y_6 \\ y_7 \end{array} \right) = \left( \begin{array}{cccc} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \\ 1&0&1&1 \\ 1&1&0&1 \\ 1&1&1&0 \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right) \ \pmod{2}. Moreover, the matrix G=\left(\begin{array}{ccccccc} 1&0&0&0&1&1&1 \\ 0&1&0&0&0&1&1 \\ 0&0&1&0&1&0&1 \\ 0&0&0&1&1&1&0 \end{array}\right) is a generator matrix.

Now, suppose a codeword c \in C is sent over a noisy channel and denote the received word by {\bf w}=(w_1,w_2,w_3,w_4,w_5,w_6,w_7). We assume at most 1 error was made.

  1. Put w_i in the region of the Venn diagram above associated to coordinate i in the table below, i=1,2,...,7.
  2. Do parity checks on each of the circles A, B, and C.

Decoding table
  parity failure region(s)     error position  
none none
A, B, and C 1
B and C 2
A and C 3
A and B 4
A 5
B 6
C 7

Exercise: Decode {\bf w} = (0,1,0,0,0,0,1) in the binary Hamming code using this algorithm.
(This is solved at the bottom of this column.)

In his 1976 autobiography Adventures of a Mathematician, Stanislaw Ulam [U] poses the following problem:

Someone thinks of a number between one and one million (which is just less than 220). Another person is allowed to ask up to twenty questions, to which the first person is supposed to answer only yes or no. Obviously, the number can be guessed by asking first: Is the number in the first half-million? and then again reduce the reservoir of numbers in the next question by one-half, and so on. Finally, the number is obtained in less than log2(1000000). Now suppose one were allowed to lie once or twice, then how many questions would one need to get the right answer?

We define Ulam’s Problem as follows: Fix integers M\geq 1 and e\geq 0. Let Player 1 choose the number from the set of integers 0,1, ...,M and Player 2 ask the yes/no questions to deduce the number, to which Player 1 is allowed to lie at most e times. Player 2 must discern which number Player 1 picked with a minimum number of questions with no feedback (in other words, all the questions must be provided at once)

As far as I know, the solution to this version of Ulam’s problem without feedback is still unsolved if e=2. In the case e=1, see [N] and [M]. (Also, [H] is a good survey.)

Player 1 has to convert his number to binary and encode it as a Hamming [7,4,3] codeword. A table containing all 16 codewords is included above for reference. Player 2 then asks seven questions of the form, “Is the i-th bit of your 7-tuple a 1?” Player 1’s answers form the new 7-tuple, (c_1,c_2,...,c_7), and each coordinate is placed in the corresponding region of the circle. If Player 1 was completely truthful, then the codeword’s parity conditions hold. This means that the 7-tuple generated by Player 2’s questions will match a 7-tuple from the codeword table above. At this point, Player 2 just has to decode his 7-tuple into an integer using the codeword table above to win the game. A single lie, however, will yield a 7-tuple unlike any in the codeword table. If this is the case, Player 2 must error-check the received codeword using the three parity check bits and the decoding table above. In other words, once Player 2 determines the position of the erroneous bit, he corrects it by “flipping its bit”. Decoding the corrected codeword will yield Player 1’s original number.

References

[H] R. Hill, “Searching with lies,” in Surveys in Combinatorics, ed. by P. Rowlinson, London Math Soc, Lecture Notes Series # 218.

[M] J. Montague, “A Solution to Ulam’s Problem with Error-correcting Codes”

[N] I. Niven, “Coding theory applied to a problem of Ulam,” Math Mag 61(1988)275-281.

[U] S. Ulam, Adventures of a mathematician, Scribner and Sons, New York, 1976 .

Solution to exercise using SAGE:


sage: MS = MatrixSpace(GF(2), 4, 7)
sage: G = MS([[1,0,0,0,1,1,1],[0,1,0,0,0,1,1],[0,0,1,0,1,0,1],[0,0,0,1,1,1,0]])
sage: C = LinearCode(G)
sage: V = VectorSpace(GF(2), 7)
sage: w = V([0,1,0,0,0,0,1])
sage: C.decode(w)
(0, 1, 0, 0, 0, 1, 1)

In other words, Player picked 4 and lied on the 6th question.

Here is a Sage subtlety:

sage: C = HammingCode(3, GF(2))
sage: C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: V = VectorSpace(GF(2), 7)
sage: w = V([0,1,0,0,0,0,1])
sage: C.decode(w)
(1, 1, 0, 0, 0, 0, 1)

Why is this decoded word different? Because the Sage Hamming code is distinct (though “equivalent” to) the Hamming code we used in the example above.

Unfortunately, SAGE does not yet do code isomorphisms (at least not as easily as GUAVA), so we use GUAVA’s CodeIsomorphism function, which calls J. Leon’s GPL’s desauto C code function:


gap> C1 := GeneratorMatCode([[1,0,0,1,0,1,0],[0,1,0,1,0,1,1],[0,0,1,1,0,0,1],[0,0,0,0,1,1,1]],GF(2));
a linear [7,4,1..3]1 code defined by generator matrix over GF(2)
gap> C2 := GeneratorMatCode([[1,0,0,0,1,1,1],[0,1,0,0,0,1,1],[0,0,1,0,1,0,1],[0,0,0,1,1,1,0]],GF(2));
a linear [7,4,1..3]1 code defined by generator matrix over GF(2)
gap> CodeIsomorphism(C1,C2);
(2,6,7,3)
gap> Display(GeneratorMat(C1));
1 . . 1 . 1 .
. 1 . 1 . 1 1
. . 1 1 . . 1
. . . . 1 1 1
gap> Display(GeneratorMat(C2));
1 . . . 1 1 1
. 1 . . . 1 1
. . 1 . 1 . 1
. . . 1 1 1 .

Now we see that the permutation (2,6,7,3) sends the “SAGE Hamming code” to the “circle Hamming code”:


sage: H = HammingCode(3, GF(2))
sage: G = MS([[1,0,0,0,1,1,1],[0,1,0,0,0,1,1],[0,0,1,0,1,0,1],[0,0,0,1,1,1,0]])
sage: C = LinearCode(G)
sage: C.gen_mat()
[1 0 0 0 1 1 1]
[0 1 0 0 0 1 1]
[0 0 1 0 1 0 1]
[0 0 0 1 1 1 0]
sage: S7 = SymmetricGroup(7)
sage: g = S7([(2,6,7,3)])
sage: H.permuted_code(g) == C
True

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