Parallel Partition and Extension:

Transcription

1 Parallel Partition and Extension: Better Permutation Arrays for Hamming Distances Sergey Bereg, Luis Gerardo Mojica, Linda Morales and Hal Sudborough Department of Computer Science, University of Texas at Dallas Richardson, Texas Abstract We give better lower bounds for M(n, d), for various positive integers d and n with d < n, where M(n, d) is the largest number of permutations on n symbols with pairwise Hamming distance at least d. Larger sets of permutations on n symbols with pairwise Hamming distance d is a necessary component of constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, Parallel Partition and Extension, is universally applicable to constructing such sets for all n and all d, d < n. I. INTRODUCTION The use of permutation codes for error correction of communications transmitted over power-lines has been suggested [1]. Due to the extreme noise in such channels, codewords are sent by frequency modulation rather than by amplitude modulation. Let s say we use frequencies f 0, f 1, f 2,..., f n 1, which we view by the index set Z n = {0, 1, 2,, n 1}. A permutation of Z n, corresponding to a codeword, specifies in which order frequencies are to be sent. The Hamming distance between two permutations on Z n, say σ and τ, denoted by hd(σ, τ), is the number of positions x in Z n such that σ(x) τ(x). For example, the permutations on Z 5, σ = and τ = have hd(σ, τ) = 3, as they differ in positions 0, 2, and 3. More information about the application can be obtained in [1], [2]. A set of permutations S on Z n (called a permutation array) has Hamming distance d, denoted by hd(s) d, if, for all σ, τ S, hd(σ, τ) d. The maximum size of a permutation array S on Z n with hd(s) d is denoted by M(n, d). There are known combinatorial upper and lower bounds on M(n, d), specifically the Gilbert-Varshamov (GV) bounds together with some recent improvements to the GV bounds [3], [4]. Generally, these bounds are theoretical and are often improved by empirical techniques. Some exact values are known: (1) for all n, M(n, n) = n, and, (2) for q that is a power of a prime, M(q, q 1) = q(q 1) and M(q+1, q 1) = (q+1)q(q 1). The exact values come from sharply k transitive groups, for k = 2 and k = 3, namely the affine general linear group, denoted by AGL, and the projective general linear group, denoted by PGL [3], [5]. The Mathieu sharply 4 and 5 transitive groups, give exact values for M(11, 8) = and M(12, 8) = [5] [7]. It is quite difficult to do a computer search for good permutation arrays on Z n when n becomes large. There are n! permutations on Z n, so the search space grows astronomically large. Some researchers have attempted to mitigate the problem by considering automorphisms groups and replacing permutations by sets of permutations. For example, in [8] Janiszczak et al considered sets of permutations invariant under isometries to improve several lower bounds for M(n, d), for various choices of n and d, but only for n 22. Chu, Colbourn and Dukes [9] and Smith and Montemanni [10] also provide lower bounds obtained by the use of automorphism groups, and are also generally limited to small values of n. There is also a connection between mutually orthogonal Latin squares (MOLS) and permutation arrays [11]. Specifically, if there are k mutually orthogonal Latin squares of side n, then M(n, n 1) kn. Let N(n) denote the number of mutually orthogonal Latin squares of side n. Finding better lower bounds for values of N(n) is an on-going combinatorial problem of considerable interest world-wide [12], [13]. Recently, in [14] we gave a new technique, called partition and extension and we illustrated how to use this technique to improve several lower bounds for M(n, n 1) over those given by MOLS. For example, suppose one has a PA A, with hd(a) n 1, and A consists of k disjoint sets (blocks), for some k, say B 1, B 2,..., B k of permutations on Z n such that: (1) for all i, hd(b i ) = n, and (2) for all i j, hd(b i, B j ) n 1. One wants to extend the PA A to a new PA A on Z n+1 by including the new symbol n, but to do so in such a way that hd(a) n. That is, one wants to increase the length of the permutations to n + 1 while simultaneously increasing the Hamming distance to n. We do this by separate partitions of: (1) the set of positions, in this case Z n, and (2) the set of symbols, in this case also Z n, each into a collection of k sets. Let s say for the positions, we use disjoint sets P 1, P 2,... P k and for the symbols we use the disjoint sets Q 1, Q 2,... Q k. We say that a permutation σ in some block B i is (c, d)-covered if some symbol d in Q i, is in some position c in P i. And, for all permutations σ in B i that are (c, d)-covered, we extend σ by putting the new symbol n in position c and putting the symbol d in position n (i.e. at the end). Observe that, as sets P i and P j in the partition of positions are disjoint, and the new symbol n is placed in blocks B i and B j in positions in the sets P i and P j, respectively, the new symbol n never appears in the same position in different blocks. So, no new agreement is created. Furthermore, in the sets Q i and Q j in the partition of symbols are disjoint, the symbols moved to position n (i.e. at the end) of blocks Bi

2 and Bj are never the same. So, again, no new agreement is created. Originally, there was at most one agreement between permutations in different blocks, so there is still at most one new agreement. As the Hamming distance is equal to the length of the permutation minus the number of agreements, and the length has increased to n+1, the inter-block Hamming distance is now (n + 1) 1 = n. Similarly, we have created at most one agreement between different permutations inside a block, whereas originally there were no agreements. So, the inter-block Hamming distance is (n + 1) 1 = n. That is, we create from a PA A for M(n, n 1) a new permutation array A for M(n + 1, n). The process is illustrated in Table I-A, where we show four cosets in the group AGL(1, 3 2 ). Notice that each of the cosets (blocks) has Hamming distance 9 and the Hamming distance between permutations in different cosets is 8. We choose the partition P = {P 0 = {0, 2, 7}, P 1 = {1, 3, 8}, P 2 = {4, 5, 6}} for the positions in cosets 0, 1 and 2 and the partition Q = {Q 0 = {1, 6, 7}, Q 1 = {3, 4, 5}, Q 2 = {0, 2, 8}} for the symbols of cosets 0, 1 and 2, respectively. The last coset, say coset 3, does not have a corresponding partition, as the new symbol, namely 9, will simply be added to the end of each permutation in that coset. The corresponding extended permutations are shown in Table I-B, where two of the permutations in coset 2 are not covered and, hence, are not included. More information about the partition and extension technique as well as many examples can be found in [14]. The purpose of this paper is to illustrate the use of partition and extension in many new ways, including from blocks defined by different Latin squares that are mutually orthogonal and between blocks defined by cosets of the cyclic subgroup of the group AGL(1, n). We describe an improvement, for example, made by an iterative approach toward defining the sets in the partition. Another new direction is to introduce several new symbols simultaneously, when appropriate, which we call parallel partition and extension. This paper illustrates several new techniques that use partition and extension to improve lower bounds for M(n, d) for various n and d. We describe new methods for creating and improving permutation arrays based on blocks defined by the cyclic subgroup C n on n symbols, which is a subgroup of AGL(1, n), and blocks defined by subsets of AGL(1, n), which are cosets of C n. These results are an improvement of results already reported in [14]. We describe an iterative technique for increasing the size of permutation arrays. In Section II, we describe two new approaches for partition and extension. These approaches lead to a new technique called parallel partition and extension, which we describe in Section III. In Section IV we describe computational techniques for partition and extension and report new results obtained by applying these techniques. Section V summarizes our work. II. PARTITION AND EXTENSION In [14] we introduced the partition and extension technique and provided detailed examples of results and partition systems used to extend permutation arrays. However, we gave only a TABLE I PARTITION AND EXTENSION EXAMPLE. COVERED AND EXTENDED PERMUTATIONS FOR FOUR COSETS IN THE GROUP AGL(1, 3 2 ). A. COVERED PERMUTATIONS B. EXTENDED PERMUTATIONS simple method. Here we provide a greedy algorithm as well as an Integer Linear Programming encoding to find such systems. In both cases, we expect solutions that are only approximations of an optimum. Nonetheless, we give many instances where the methods result in optimum solutions. An optimum partition system is one that covers all permutations in all blocks. In general, there may not be an optimum partition system. A. Greedy Approach to Partition and Extension Formally, a partition system Π = (M, P, Q) consists of a set M of permutations on Z n, which are arranged into k + 1 blocks, say B 1, B 2,..., B k+1, a partition P of the positions for blocks B 1, B 2,..., B k+1, and a partition Q of the symbols for blocks B 1, B 2,..., B k+1. For the block B k+1, the symbol n is simply appended to the end of each permutation, so partitions P and Q need not include positions and symbols for the permutations in B k+1 [14]. Instead of exploring all possible partitions, we iteratively find the best current solution to the sub-problem in hand. In Algorithm 1 we provide a greedy algorithm that follows such a strategy and has shown to outperform trivial partitioning systems and obtained numerous new lower bounds. The high level idea is to begin with a simplification of the problem, the algorithm creates an arbitrary partition Q of symbols by evenly dividing all symbols among k blocks.

3 The algorithm then iterates to find a partition of positions P that maximizes the number of covered permutations. At every iteration, an available position p is selected. For each block B i and ist corresponding set of symbols Q i, the number of covered permutations is counted. After all bocks have been tested, position p is assigned to the block that obtains maximum coverage, with ties broken randomly. Once a position is assigned, the decision is not changed in later steps. The process repeats until all positions in Z n are assigned to blocks. Refer to Algorithm 1 for full details. Algorithm 1: Greedy Partition and Extension Algorithm Data: M, a list of blocks with d + 1 and d, intra an inter hamming distance respectively. k, the number of blocks 0 k n and n > 1, the number of symbols in Z n 1. Result: Q, P a partition of symbols and positions, respectively. 2 begin 3 Q {Q 0 = {0, 1,..., k 1}, Q 1 = {k,..., 2k 1},... Q k = {n 1,..., nk 1}} 4 P {P 0 = {}, P 1 = {},... P k = {}} 5 P A {0, 1,... n 1} /* Set of Available positions */ 6 for p P A do 7 cov [ ] /* Temporary array for tracking block coverage */ 8 for 0 i k do 9 cov[i] = coverage(m i, Q i, p) 10 m idx =argmax(cov) 11 P [midx ] P [midx ] p 12 P A P A p 14 return P 1 Procedure coverage(m i, Q i, p) 2 count 0 3 for perm M i do 4 if perm[p] Q i then 5 count count return count B. An optimization approach to Partition and Extension The greedy approach for finding a partition system described in Section II-A simplifies the problem by arbitrarily creating Q, the partition of symbols in Z n, and greedily creating the partition of positions for P. The benefit of such changes is the reduction of running time and search space, at the cost of possibly missing an optimal solution. In order to outperform the greedy algorithm results, one can expand the search space and explore a larger set of solutions. Since the possibility of exhaustively exploring all possible partitions of symbols and positions remains unattainable, we still arbitrarily create the partition symbols Q, but encode the search for the partition of positions P as an optimization problem. We cast the search for P as an ILP problem and allow an off-the-shelf solver to explore the entire search space of partitions for P. There exist commercial solvers [15], [16], capable of solving large ILP problems efficiently, like the one described in this section. Another advantage of this approach is that the same ILP formulation is its flexibility, it allows the user to decide the amount of resources ( computation time, processors, etc.) allocated to the problem, as these parameters are usually tunable in the solver. It is important to recall that despite of these advantages, ILP is a NP-Hard problem [17], and its solution may be only an approximation to the optimum. We now describe an ILP encoding of the search for a partition of positions P, given a partition of symbols Q and a set of M permutations arranged in k blocks. Let c i,j be a binary variable indicating that permutation j of block i is covered. Let u(i) be a function that maps the block index i to the number of permutations in it. Also, let b i,p be a binary variable indicating that position p is assigned to block i. where maximize c i,j subject to k 1 i=0 u(i) 1 j=0 c i,j (1) k 1 b i,p = 1; p; (2) i=0 y Q i 1 σp,y b i,p c i,j ; i, j, p; (3) b i,p = n; (4) k 1 n 1 i=0 p=0 1 σp,y = { 1 if σ[p] = y 0 otherwise. Equation (1) is the objective function to be maximized, that is, the total number of permutations covered in all blocks in M. The optimization is subject to three constraints: Equation (2) assures that the resulting partition P assigns a position to exactly one block, Equation (3) establishes that permutation j in block i is covered when the at least one of its symbols listed in Q i appears in position p, and p is assigned to this block i, and Equation (4) enforces the solution to assign each and every position to a block. Constraints 2 and 4 effectively ensure that the solution is a partition. Equation (5) defines an indicator function that states whether or not a permutation σ is covered by checking if symbol y appears at position p. III. PARALLEL PARTITION AND EXTENSION In Section II we provided two different methods for finding partition systems based on the partition and extension technique [14]. Recall that the partition and extension technique extends a permutation array by swapping one new symbol with an existing symbols in a carefully selected position. The (5)

4 displaced symbol is appended at the end of the permutation. Parallel partition and extension introduces one or more symbols simultaneously when appropriate. A permutation array obtained from parallel partition is potentially larger than one obtained from partition and extension iteratively. A. Rudimentary Parallel Partition and Extension A rudimentary form of parallel partition and extension was described in [18], where a PA A on Z n was extended to a PA A on Z n+r by introducing, to each permutation in A, r new symbols simultaneously. Specifically, suppose A has k = 2r blocks B 0, B 1,..., B k 1, for some r, with, for all i, hd(b i ) d, for some d, and, for all i j, hd(b i, B j ) d r. In particular, hd(a) d r, and a new PA A is created such that hd(a ) d. The sequence γ of new symbols, namely γ = n, n+1,..., n+r 1, is appended to each permutation in each of the blocks B 0, B 1,..., B k 1, but the order and position of the sequence in different blocks is changed by a cyclic shift. In particular, for each integer t (0 t < r), let shift (γ, t) denote the left cyclic shift of the sequence by t positions. For example, shift (γ, 1) is the sequence n + 1, n + 2,..., n + r 1, n and shift (γ, 2) is the sequence n + 2,..., n + r 1, n, n + 1. Then, in block B i, for all i (0 i < r), the first r symbols in each permutation of B i, are replaced by shift(γ, i), and the r replaced symbols are put in their original order at the end of the permutation in positions n, n + 1,..., n + r 1. And, in block B i, for all i (r i < 2r), the sequence shift (γ, i) is added to each permutation in positions n, n+1,..., n+r 1. It is known that the Hamming distance between two permutations does not change when the order of the symbols in both permutations is altered in a fixed manner. Consequently, the movement of the first r symbols in each permutation to positions n, n+1,..., n+r 1 does not change the Hamming distance between permutations in the same block or between permutations in different blocks. Since the ordering of the new symbols n, n+1,..., n+r 1 is different in different blocks, no new agreement between permutations in different blocks is created. For the original permutation array A, hd(a) d r. For the new permutation array A the permutations in each blocks have been extended by r symbols arranged in a way that ensures that the intra-block Hamming distance is d, then the length of the permutations has increased by r. Within each block, the r new symbols are put in a fixed order into a fixed positions, thus creating r new agreements, and the displaced symbols were moved to the end of each permutation. Thus the intra-block Hamming distance is d. Hence, hd(a ) d, and A is made of covered permutations from k = 2r blocks. B. A Better Parallel Partition and Extension We now provide a richer version of the parallel partition and extension technique. The motivation for this technique is the observation that the rudimentary version introduces new symbols in the first r positions of the permutations in a permutation array A. That means that there is possibly a large number of positions in addition to the first r that can be used to introduce new symbols allowing the extension of a larger number of blocks. In order to use a larger number of positions without creating agreements between permutations in different blocks, we use the partition and extension technique described in Section II to obtain a partition system and introduce new symbols based on its P and Q. Let (M, P and Q) be a partition system. In standard partition and extension, M consists of k + 1 blocks, say B 0, B 1,..., B k of permutations (the last block need not be partitioned), P is a partition of the positions Z n, into k disjoint sets, and Q is a partition of the symbols, i.e. Z n, into k disjoint sets. We illustrate the richer for of parallel partition and extension, where r new symbols are to be added simultaneously. For example, we can use a partition system (M, P and Q), where M consists of v blocks, where v k + r, say B 0, B 1,..., B v 1, and, where, for all i, hd(b i ) d, and, for all i j, hd(b i, B j ) d r. As before, P is a partition of the positions, Z n, into k disjoint sets, and Q is a partition of the symbols Z n, into k disjoint sets. For example, let s assume that k = 4, r = 3 and v = 7. So, P is a partition of Z n into four disjoint subsets (of positions), say P 0, P 1, P 2, and P 3 and Q is a partition of Z n into four disjoint subsets (of symbols), say Q 0, Q 1, Q 2, and Q 3. Then, for all i (0 i 3), permutations in B i are covered if they have in some position of each of the sets P i, P i+1( mod 4), P i+2( mod 4) at least one symbol in each of the sets Q i, Q i+1( mod 4), Q i+2( mod 4). The blocks B 0, B 1, B 2, B 3 are extended as follows: for a permutation σ in B i (a) replace a symbol in Q i the first time it occurs in a position of P i with the new symbol n, (b) replace a symbol in Q i+1( mod 4) the first time it occurs in a position of Q i+1( mod 4) with the new symbol n + 1, and (c) replace a symbol in Q i+2( mod 4) the first time it occurs in a position of P i+2( mod 4) with the new symbol n + 2. This extends four of the seven blocks. The last three blocks, B 4, B 5 and B 6 can be extended by inserting at the end of each block a different cyclic shift of the sequence of new symbols n, n + 1, n + 2. Altogether, we transform the PA A on Z n, consisting of the blocks B 0, B 1, B 2, B 3, B 4, B 5, B 6, with hd(a) d r into a PA A on Z n+4 with hd(a ) d. This is better than rudimentary parallel partition and extension, as the rudimentary technique allows at most six blocks to be extended. That is, a permutation array created by this technique contains more permutations than a similar permutation array created by the rudimentary technique. Clearly, the more general form of parallel partition and extension will work for other values of k, r, and v. A more detailed and complete description of our improved parallel partition and extension technique will be given in a forthcoming paper. Space restrictions for the current publication make such details infeasible. Example. In Table II we give six blocks in the group AGL(1, 9). Inside each block the Hamming distance is 9 and between two blocks the Hamming distance is 8. We use this as an example of the technique, as it is small, even though when adding three new symbols, normally, the Hamming distance inside each block should be three greater than the Hamming

5 distance between permutations in different blocks. We choose the partition of the positions to be P = {P 0 = {0, 2, 7}, P 1 = {1, 3, 8}, P 2 = {4, 5, 6}} and the partition of the symbols to be Q = {Q 0 = {1, 6, 7}, Q 1 = {2, 4, 8}, Q 2 = {0, 3, 5}}. Then the first permutation in the first block is covered, because (a) it has in position 7 in set P 0 the symbol 7 in set Q 0, (b) it has in position 8 in P 1 the symbol 8 in set Q 1, and (c) it has in position 5 in P 2 the symbol 5 in set Q 2. Similarly, the second permutation in the first block is covered, because (a) it has in position 0 in set P 0 the symbol 1 in set Q 0, (b) it has in position 3 in P 1 the symbol 4 in set Q 1, and (c) it has in position 5 in P 2 the symbol 0 in set Q 2. The third permutation in the first block is covered, because (a) it has in position 2 in set P 0 the symbol 6 in set Q 0, (b) it has in position 1 in P 1 the symbol 8 in set Q 1, and (c) it has in position 4 in P 2 the symbol 5 in set Q 2. However, the fourth permutation in block four is not covered, because no symbol in Q 2 = {0, 2, 8} is in any of the positions in P 2 = {4, 5, 6}. The rest of the permutations covered are shown in Table II. It should be noted that in block two, one makes a cyclic shift of the sets of symbols, so we say a permutation is covered if it (a) has a symbol in set Q 1 = {3, 4, 5} in one of the positions P 0 = {0, 2, 7}, (b) has a symbol in set Q 2 = {0, 2, 8} in one of the positions P 1 = {1, 3, 8}, and (c) has a symbol in set Q 0 = {1, 6, 7} in one of the positions P 2 = {4, 5, 6}. Similarly, in block three, one makes another cyclic shift, so a permutation is covered if it (a) has a symbol in set Q 2 = {0, 2, 8} in one of the positions P 0 = {0, 2, 7}, (b) has a symbol in set Q 0 = {1, 6, 7} in one of the positions P 1 = {1, 3, 8}, and (c) has a symbol in set Q 1 = {3, 4, 5} in one of the positions P 2 = {4, 5, 6}. IV. EXPERIMENTS AND RESULTS As mentioned in section II, our greedy and ILP formulations produced a number of new bounds. A subset of them for M(n, n 1) is given Table III. This table specifies the number of symbols in Z n, the previously known best bound, the new bound and the method used to obtained it: ILP or the greedy algorithm. Some results were previously published in the paper referenced in the MET column. We observed that as the number of symbols in Z n increased, both time and space increased for the ILP method. There were times the greedy method was incapable of finding solutions as good as those found with ILP. Table IV reports new bounds obtained from the parallel partition and extension technique described in section III. We experimented with a number of blocks k to be extended and report results obtained from the best k. For the entries in table IV where the number of new symbols is 1, the parallel partition and extension technique reduces to partition and extension. When this occurs we report results obtained using Algorithm 1 or the ILP solver. All experiments were conducted in a Intel Xeon computer with four cores and 8 gigabytes of RAM. The resulting permutation arrays, as well as their partitions files can be found in [19]. TABLE II PARALLEL PARTITION AND EXTENSION EXAMPLE. COVERED AND EXTENDED PERMUTATIONS (WITH THREE NEW SYMBOLS) FOR FOUR COSETS IN THE GROUP AGL(1, 3 2 ). A. COVERED PERMUTATIONS B. EXTENDED PERMUTATIONS V. CONCLUSION We presented two methods for the partition and extension technique that produced a number of competitive new bounds. Also, we presented a richer version of parallel partition and extension that introduces one or more symbols simultaneously and, when conditions apply, it outperforms standard parallel partition and extension and sequential application of the partition and extension technique and is universally applicable. We report experimental results and provide tables with new lower bounds for M(n, n 1) and M(n, d).

6 TABLE III M(n, n 1) LOWER BOUNDS. N DENOTES THE NUMBER OF SYMBOLS IN THE PERMUTATION ARRAY, PRV THE PREVIOUSLY KNOWN BOUND, NEW THE NEW BOUND AND MET THE METHOD USED TO OBTAIN THE NEW BOUND, ILP STANDS FOR THE ILP FORMULATION AND GRD INDICATED THE GREEDY ALGORITHM. N PRV NEW MET N PRV NEW MET grd 270 1,890 4,318 [14] ilp 272 4,080 4,408 [14] ilp 274 1,644 3,706 grd ilp 276 2,760 3,575 ilp ilp 278 1,668 4,574 [14] ilp 280 2,232 2,511 grd ilp 282 4,684 4,777 ilp ilp 284 4,706 4,811 ilp grd 286 2,860 3,420 ilp grd 294 1,764 5,068 [14] grd 306 1,836 4,575 ilp 108 1,090 1,167 ilp 308 2,156 5,360 [14] 110 1,130 1,188 ilp 316 2,212 3,150 grd grd 322 1,932 4,815 ilp ,293 ilp 330 1,980 2,961 grd grd 335 2,010 2,338 grd 132 1,508 1,563 ilp 340 2,040 2,373 grd grd 346 2,076 2,415 grd 138 1,614 1,644 ilp 356 2,492 3,195 grd 140 1,640 1,721 ilp 358 2,148 3,213 grd grd 366 2,196 2,555 grd 145 1,152 1,422 ilp 370 2,952 5,535 ilp 146 1,010 1,015 grd 372 2,604 5,565 ilp 148 1,027 1,029 grd 376 2,632 5,625 ilp 150 1,818 1,902 ilp 378 4,524 4,901 ilp 152 1,832 1,943 ilp 382 2,674 4,572 ilp 155 1,207 1,232 grd 386 2,702 5,775 ilp ,085 grd 388 3,096 5,805 ilp 158 1,922 2,052 ilp 404 4,836 6,045 ilp ,106 grd 405 3,240 4,444 grd 161 1,377 1,440 ilp 406 2,842 3,240 grd ,127 grd 408 4,070 6,105 ilp 164 2,042 2,193 ilp 412 3,296 5,343 grd 166 1,153 1,155 grd 414 4,140 4,956 grd 168 2,070 2,231 ilp 415 3,735 4,140 grd 177 1,760 2,113 grd 417 6,255 6,924 grd 178 1,068 1,593 ilp 418 2,926 6,255 ilp 180 2,404 2,529 ilp 424 3,384 6,345 ilp 182 1,092 2,498 [14] 430 2,580 3,003 grd 186 1,619 1,665 grd 436 2,616 6,525 ilp 190 1,499 1,512 grd 442 3,528 6,615 ilp 194 2,680 2,702 ilp 446 3,122 5,785 ilp 196 1,176 1,365 grd 452 4,510 6,765 ilp 204 1,420 1,421 grd 456 3,192 6,825 ilp 206 1,638 1,640 grd 466 3,262 6,975 ilp 209 2,496 2,912 ilp 470 3,290 3,752 grd 210 2,100 2,299 ilp 472 3,304 7,065 ilp 212 3,026 3,136 ilp 474 4,740 7,095 ilp 214 1,284 1,491 grd 478 3,816 7,155 ilp 218 1,308 1,736 grd 482 5,772 7,215 ilp 220 2,099 2,190 ilp 484 3,872 7,245 ilp 224 3,260 3,345 ilp 485 3,395 3,872 grd 225 1,800 2,743 grd 486 2,916 3,395 grd 228 1,368 3,380 [14] 490 2,940 7,335 grd 230 1,380 3,512 [14] 498 2,988 7,455 ilp 234 1,404 3,602 [14] 506 3,036 7,575 ilp 236 1,416 1,645 grd 508 3,556 7,605 ilp 238 1,428 1,659 grd 516 4,128 7,725 grd 240 1,680 3,656 [14] 518 5,170 6,204 grd 242 3,716 3,817 ilp 520 4,160 7,785 grd 244 1,464 3,116 ilp 526 4,208 7,875 ilp 246 1,476 1,715 grd 532 4,256 7,965 ilp 250 1,500 1,743 grd 536 4,288 8,025 ilp 252 3,932 4,016 ilp 538 5,380 8,055 ilp 254 2,530 3,023 ilp 540 6,480 8,085 ilp 255 1,785 2,286 grd 545 8,704 9,792 grd 258 1,548 4,066 [14] 561 3,927 8,400 ilp 260 1,560 3,108 ilp 566 3,396 3,955 grd 264 1,848 4,228 [14] 582 4,074 4,648 grd 266 1,862 2,120 grd 586 4,102 4,680 grd 268 1,876 2,670 grd 591 4,137 10,030 ilp TABLE IV M(n, d) LOWER BOUNDS. 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