2 Bounds on the size of codes
2.7 The Griesmer Upper Bound
The final upper bound we discuss is a generalization of the Singleton Bound known as the Griesmer Bound. We place it last because, unlike our other upper bounds, this one applies only to linear codes.
To prove this bound we first discuss the generally useful idea of a residual code due to H. J. Helgert and R. D. Stinaff [120]. LetCbe an [n,k] code and letcbe a codeword of weightw. Let the set of coordinates on whichcis nonzero beI. Then theresidual code ofCwith respect toc, denoted Res(C,c), is the code of lengthn−wpunctured on all the coordinates ofI. The next result gives a lower bound for the minimum distance of residual codes [131].
Theorem 2.7.1 LetCbe an[n,k,d]code overFqand letcbe a codeword of weightw <
(q/(q−1))d. ThenRes(C,c)is an[n−w,k−1,d]code,where d≥d−w+ "w/q#.
Proof: By replacing C by a monomially equivalent code, we may assume that c= 11· · ·100· · ·0. Since puncturing c on its nonzero coordinates gives the zero vector, Res(C,c) has dimension less thank. Assume that the dimension is strictly less thank−1.
Then there exists a nonzero codewordx=x1· · ·xn∈Cwhich is not a multiple ofcwith xw+1· · ·xn=0. There existsα∈Fq such that at leastw/q coordinates ofx1· · ·xwequal α. Therefore
d ≤wt(x−αc)≤w−w
q = w(q−1)
q ,
contradicting our assumption onw. Hence Res(C,c) has dimensionk−1.
81 2.7 The Griesmer Upper Bound
We now establish the lower bound ford. Letxw+1· · ·xn be any nonzero codeword in Res(C,c), and letx=x1· · ·xwxw+1· · ·xnbe a corresponding codeword inC. There exists α∈Fq such that at leastw/q coordinates ofx1· · ·xwequalα. So
d≤wt(x−αc)≤w−w
q +wt(xw+1· · ·xn).
Thus every nonzero codeword of Res(C,c) has weight at leastd−w+ "w/q#. Applying Theorem 2.7.1 to a codeword of minimum weight we obtain the following.
Corollary 2.7.2 IfCis an[n,k,d]code overFqandc∈Chas weight d,thenRes(C,c)is an[n−d,k−1,d]code,where d≥ "d/q#.
Recall that the Nordstrom–Robinson code defined in Section 2.3.4 is a nonlinear binary code of length 16 and minimum distance 6, with 256=28codewords, and, as we described, its existence together with the Johnson Bound (see Example 2.3.9) implies thatA2(16,6)= 28. It is natural to ask whetherB2(16,6) also equals 28. In the next example we illustrate how residual codes can be used to show that no [16,8,6] binary linear code exists, thus implying thatB2(16,6)≤27.
Example 2.7.3 LetCbe a [16,8,6] binary linear code. LetC1be the residual code ofC with respect to a weight 6 vector. By Corollary 2.7.2,C1is a [10,7,d] code with 3≤d; by the Singleton Boundd≤4. Ifd=4,C1 is a nontrivial binary MDS code, which is impossible by Theorem 2.4.4. Sod=3. Notice that we have now reduced the problem to showing the nonexistence of a [10,7,3] code. But the nonexistence of this code follows from the Sphere Packing Bound as
27> 210 10
0
+ 10
1
.
Exercise 120 In Exercise 93, we showed thatB2(13,6)≤25. Show using residual codes thatB2(13,6)≤24. Also construct a code that meets this bound.
Exercise 121 Do the following:
(a) Use the residual code to prove that a [16,5,8] binary code contains the all-one code-word1.
(b) Prove that a [16,5,8] binary code has weight distribution A0= A16=1 andA8=30.
(c) Prove that all [16,5,8] binary codes are equivalent.
(d) Prove thatR(1,4) is a [16,5,8] binary code.
Theorem 2.7.4 (Griesmer Bound[112]) LetC be an[n,k,d]code overFq with k ≥1.
Then n≥
k−1
i=0
"
d qi
# .
Proof: The proof is by induction onk. Ifk=1 the conclusion clearly holds. Now assume thatk>1 and letc∈C be a codeword of weightd. By Corollary 2.7.2, Res(C,c) is an [n−d,k−1,d] code, whered≥ "d/q#. Applying the inductive assumption to Res(C,c), we haven−d ≥k−2
i=0"d/qi+1#and the result follows.
Since"d/q0# =dand"d/qi# ≥1 fori =1, . . . ,k−1, the Griesmer Bound implies the linear case of the Singleton Bound.
The Griesmer Bound gives a lower bound on the length of a code overFqwith a prescribed dimensionkand minimum distanced. The Griesmer Bound does provide an upper bound onBq(n,d) because, givennandd, there is a largestkfor which the Griesmer Bound holds.
ThenBq(n,d)≤qk.
Givenk,d, andqthere need not exist an [n,k,d] code overFqwhich meets the Griesmer Bound; that is, no code may exist where there is equality in the Griesmer Bound. For example, by the Griesmer Bound, a binary code of dimensionk=12 and minimum distance d =7 has lengthn≥22. Thus the [23,12,7] binary Golay code does not meet the Griesmer Bound. But a [22,12,7] binary code does not exist because the Johnson Bound (2.13) gives
A2(22,7)≤ 222/2025 =2071, implyingB2(22,7)≤211.
It is natural to try to construct codes that meet the Griesmer Bound. We saw that the [23,12,7] binary Golay code does not meet the Griesmer Bound. Neither does the [24,12,8]
extended binary Golay code, but both the [12,6,6] and [11,6,5] ternary Golay codes do.
(See Exercise 122.) In the next theorem, we show that the [(qr−1)/(q−1),r] simplex code meets the Griesmer Bound; we also show that all its nonzero codewords have weight qr−1, a fact we verified in Section 1.8 whenq =2.
Theorem 2.7.5 Every nonzero codeword of the r -dimensional simplex code overFq has weight qr−1. The simplex codes meet the Griesmer Bound.
Proof: LetG be a generator matrix for ther-dimensional simplex codeC overFq. The matrixGis formed by choosing for its columns a nonzero vector from each 1-dimensional subspace ofFrq. BecauseC= {xG|x∈Frq}, ifx=0, then wt(xG)=n−s, wheresis the number of columnsyofGsuch thatx·yT=0. The set of vectors ofFrqorthogonal toxis an (r−1)-dimensional subspace ofFrqand thus exactly (qr−1−1)/(q−1) columnsyofG satisfyx·yT=0. Thus wt(xG)=(qr−1)/(q−1)−(qr−1−1)/(q−1)=qr−1proving that each nonzero codeword has weightqr−1.
In particular, the minimum distance isqr−1. Since
r−1
i=0
"
qr−1 qi
#
=
r−1
i=0
qi =(qr−1)/(q−1),
the simplex codes meet the Griesmer Bound.
Exercise 122 Prove that the [11,6,5] and [12,6,6] ternary Golay codes meet the Griesmer Bound, but that the [24,12,8] extended binary Golay code does not.
Solomon and Stiffler [319] and Belov [15] each construct a family of codes containing the simplex codes which meet the Griesmer Bound. Helleseth [122] has shown that in
83 2.7 The Griesmer Upper Bound
many cases there are no other binary codes meeting the bound. For nonbinary fields, the situation is much more complex. Projective geometries have also been used to construct codes meeting the Griesmer Bound (see [114]).
In general, an [n,k,d] code may not have a basis of minimum weight vectors. However, in the binary case, if the code meets the Griesmer Bound, it has such a generator matrix, as the following result of van Tilborg [328] shows.
Theorem 2.7.6 LetCbe an[n,k,d]binary code that meets the Griesmer Bound. ThenC has a basis of minimum weight codewords.
Proof: We proceed by induction onk. The result is clearly true ifk=1. Assume thatcis a codeword of weightd. By permuting coordinates, we may assume thatChas a generator matrix the Greismer Bound. By induction, we may assume the rows ofG1 have weightd1. For i ≥2, letri =(si−1,ti−1) be rowiofG, wheresi−1is rowi−1 ofG0andti−1is rowi−1 ofG1. By Exercise 124, one ofri orc+ri has weightd. HenceChas a basis of weightd
codewords.
Exercise 123 Prove that fori ≥1,
"
Exercise 124 In the notation of the proof of Theorem 2.7.6 show that one ofri orc+ri
has weightd.
Exercise 125 Prove that ifdis even, a binary code meeting the Griesmer Bound has only
even weight codewords. Do not use Theorem 2.7.9.
This result has been generalized by Dodunekov and Manev [70]. Let g(k,d)= be the summation in the binary Griesmer Bound. The Griesmer Bound says that for an [n,k,d] binary code to exist,n≥g(k,d). So n−g(k,d) is a measure of how close the
length of the code is to one that meets the Griesmer Bound. It also turns out to be a measure of how much larger than minimum weight the weights of your basis vectors may need to be.
Theorem 2.7.7 LetCbe an[n,k,d]binary code with h=n−g(k,d). ThenChas a basis of codewords of weight at most d+h.
Proof: We proceed by induction onh. The caseh=0 is covered by Theorem 2.7.6. For fixedh, proceed by induction onk. In the casek=1, there certainly is a basis of one codeword of weightd. Whenk>1 assume thatcis a codeword of weightd. By permuting coordinates, we may assume thatChas a generator matrix
G=
Exercise 126 Let x and y be nonnegative real numbers. Show that "x+y# ≥ "x# +
y.
Exercise 127 Show thatg(k−1,"d/2#)=g(k,d)−d. Exercise 128 In the notation of the proof of Theorem 2.7.7 show that one ofrj orc+rj
has weight betweendandd+h.
85 2.7 The Griesmer Upper Bound
Exercise 129
(a) Computeg(5,4) from (2.40).
(b) What are the smallest weights that Theorem 2.7.7 guarantees can be used in a basis of a [10,5,4] binary code?
(c) Show that a [10,5,4] binary code with only even weight codewords has a basis of weight 4 codewords.
(d) Construct a [10,5,4] binary code with only even weight codewords.
Both Theorems 2.7.6 and 2.7.7 can be generalized in the obvious way to codes over Fq; see [68]. In particular, codes overFq that meet the Griesmer Bound have a basis of minimum weight codewords. This is not true of codes in general but is true for at least one code with the same parameters, as the following theorem of Simonis [308] shows.
This result may prove to be useful in showing the nonexistence of linear codes with given parameters [n,k,d].
Theorem 2.7.8 Suppose that there exists an[n,k,d]codeCoverFq. Then there exists an [n,k,d]codeCwith a basis of codewords of weight d.
Proof: Letsbe the maximum number of independent codewords{c1, . . . ,cs}inCof weight d. Note thats≥1 asChas minimum weightd. We are done ifs=k. So assumes<k.
The theorem will follow by induction if we show that we can create fromCan [n,k,d]
codeC1with at leasts+1 independent codewords of weightd. LetS =span{c1, . . . ,cs}.
By the maximality ofs, every vector inC\Shas weight greater thand. Lete1be a min-imum weight vector inC\S with wt(e1)=d1 >d. Complete{c1, . . . ,cs,e1}to a basis {c1, . . . ,cs,e1,e2, . . . ,ek−s}ofC. Choosed1−d nonzero coordinates ofe1and createe1 to be the same ase1except on thesed1−d coordinates, where it is 0. So wt(e1)=d. Let C1=span{c1, . . . ,cs,e1,e2, . . . ,ek−s}. We showC1 has minimum weightd and dimen-sionk. The vectors inC1fall into two disjoint sets:S=span{c1, . . . ,cs}andC1\S. The nonzero codewords inS have weightd or more asS ⊂C. The codewords inC1\S are obtained from those inC\S by modifying d1−d coordinates; therefore as C\S has minimum weightd1,C1\S has minimum weight at leastd. SoC1has minimum weight d. Suppose that C1 has dimension less than k. Then by our construction, e1 must be in span{c1, . . . , cs,e2, . . . ,ek−s} ⊂C. By maximality of s,e1 must in fact be in S. So e1−e1∈C\S as e1∈S. By construction wt(e1−e1)=d1−d; on the other hand as e1−e1∈C\S, wt(e1−e1)≥d1, sinced1is the minimum weight ofC\S. This contra-diction shows thatC1 is an [n,k,d] code with at leasts+1 independent codewords of
weightd.
Exercise 130 LetCbe the binary [9,4] code with generator matrix
1 0 0 0 1 1 1 0 0
0 1 0 0 1 1 0 1 0
0 0 1 0 1 0 1 1 1
0 0 0 1 0 1 1 1 1
.
(a) Find the weight distribution ofCand show that the minimum weight ofCis 4.
(b) Apply the technique of Theorem 2.7.8 to construct a [9,4,4] code with a basis of weight 4 vectors.
(c) Choose any three independent weight 4 vectors inC and any weight 5 vector inC. Modify the latter vector by changing one of its 1s to 0. Show that these four weight 4
vectors always generate a [9,4,4] code.
Exercise 125 shows that a binary code meeting the Griesmer Bound has only even weight codewords ifdis even. The next theorem extends this result. The binary case is due to Dodunekov and Manev [69] and the nonbinary case is due to Ward [346].
Theorem 2.7.9 Let C be a linear code over Fp, where p is a prime, which meets the Griesmer Bound. Assume that pi |d. Then pi divides the weights of all codewords ofC; that is,pi is a divisor ofC.