<p>You are given an array <code>nums</code>​​​ and an integer <code>k</code>​​​​​. The <font face="monospace">XOR</font> of a segment <code>[left, right]</code> where <code>left &lt;= right</code> is the <code>XOR</code> of all the elements with indices between <code>left</code> and <code>right</code>, inclusive: <code>nums[left] XOR nums[left+1] XOR ... XOR nums[right]</code>.</p>

<p>Return <em>the minimum number of elements to change in the array </em>such that the <code>XOR</code> of all segments of size <code>k</code>​​​​​​ is equal to zero.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,0,3,0], k = 1
<strong>Output:</strong> 3
<strong>Explanation: </strong>Modify the array from [<u><strong>1</strong></u>,<u><strong>2</strong></u>,0,<u><strong>3</strong></u>,0] to from [<u><strong>0</strong></u>,<u><strong>0</strong></u>,0,<u><strong>0</strong></u>,0].
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [3,4,5,2,1,7,3,4,7], k = 3
<strong>Output:</strong> 3
<strong>Explanation: </strong>Modify the array from [3,4,<strong><u>5</u></strong>,<strong><u>2</u></strong>,<strong><u>1</u></strong>,7,3,4,7] to [3,4,<strong><u>7</u></strong>,<strong><u>3</u></strong>,<strong><u>4</u></strong>,7,3,4,7].
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,4,1,2,5,1,2,6], k = 3
<strong>Output:</strong> 3
<strong>Explanation: </strong>Modify the array from [1,2,<strong><u>4,</u></strong>1,2,<strong><u>5</u></strong>,1,2,<strong><u>6</u></strong>] to [1,2,<strong><u>3</u></strong>,1,2,<strong><u>3</u></strong>,1,2,<strong><u>3</u></strong>].</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 2000</code></li>
	<li><code>​​​​​​0 &lt;= nums[i] &lt; 2<sup>10</sup></code></li>
</ul>

# from typing import List
# from collections import defaultdict

# class Solution:
#     def minChanges(self, nums: List[int], k: int) -> int:
#         n = len(nums)
#         max_val = 2 ** 10  # The maximum possible XOR value

#         # Initialize a dynamic programming array to store minimum changes for each XOR value
#         dp = [float('inf')] * max_val
#         dp[0] = 0  # Base case: No changes needed for XOR 0

#         # Iterate over each position in the subarray of size k
#         for i in range(k):
#             freq = defaultdict(int)  # Dictionary to store frequency of elements in the current subarray
#             count = 0  # Count of elements in the current subarray
#             # Iterate over the subarray with a step of k
#             for j in range(i, n, k):
#                 freq[nums[j]] += 1
#                 count += 1

#             prev_dp = min(dp)  # Get the minimum value from the previous dp array

#             new_dp = [prev_dp + count] * max_val  # Initialize a new dp array
#             # Combine loops over possible XOR values and elements in the current subarray
#             for xor in range(max_val):
#                 for num, freq_num in freq.items():
#                     new_xor = xor ^ num  # Calculate the new XOR value
#                     # Update the dp array with the minimum changes needed
#                     new_dp[new_xor] = min(new_dp[new_xor], dp[xor] - freq_num + count)

#             dp = new_dp  # Update dp array for the next position in the subarray

#         return dp[0]  # Return the minimum changes needed for XOR 0

from typing import List
from collections import defaultdict
import functools

class Solution:
    def minChanges(self, nums: List[int], k: int) -> int:
        max_xor = 1024  # 2^10
        
        # Initialize dp table
        dp = [float('inf')] * max_xor
        dp[0] = 0  # No changes needed to get XOR of 0 initially
        
        for i in range(k):
            # Frequency map for current group (i-th position in each block)
            freq = defaultdict(int)
            total = 0
            for j in range(i, len(nums), k):
                freq[nums[j]] += 1
                total += 1
            
            # Find the minimum changes for this block
            new_dp = [min(dp)] * max_xor  # Start with minimum changes possible for the whole previous dp state
            for xor_val in range(max_xor):
                for num, count in freq.items():
                    new_dp[xor_val] = min(new_dp[xor_val], dp[xor_val ^ num] - count)
            
            for j in range(max_xor):
                new_dp[j] += total
            
            dp = new_dp
        
        return dp[0]
3204